CHAPTER 4�VECTOR SPACES

4.1 VECTORS IN R^N

4.2 VECTOR SPACES

4.3 SUBSPACES OF VECTOR SPACES

4.4 SPANNING SETS AND LINEAR INDEPENDENCE

4.5 BASIS AND DIMENSION

4.6 RANK OF A MATRIX AND SYSTEMS OF LINEAR EQUATIONS

4.7 COORDINATES AND CHANGE OF BASIS

4.8 APPLICATIONS OF VECTOR SPACES

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Elementary Linear Algebra

R. Larson (8 Edition)

投影片設計製作者

淡江大學 電機系 翁慶昌 教授

CH 4 Linear Algebra Applied

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Force (p.157) Image Morphing (p.180)

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Digital Sampling (p.172)

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Crystallography (p.207) Satellite Dish (p.217)

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4.1 Vectors in Rⁿ

a sequence of n real number

• An ordered n-tuple:

Elementary Linear Algebra: Section 4.1, p.155

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• Ex:

Elementary Linear Algebra: Section 4.1, p.155

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• Notes:

• Ex:

(1) An n-tuple can be viewed as a point in Rⁿ with the x_i’s as its coordinates.

(2) An n-tuple can be viewed as a vector

in Rⁿ with the x_i’s as its components.

Elementary Linear Algebra: Section 4.1, p.152

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(two vectors in Rⁿ)

• Equal:

if and only if

• Vector addition (the sum of u and v):

• Scalar multiplication (the scalar multiple of u by c):

• Notes:� The sum of two vectors and the scalar multiple of a vector

in Rⁿ are called the standard operations in Rⁿ.

Elementary Linear Algebra: Section 4.1, p.153

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• Negative:

Elementary Linear Algebra: Section 4.1, p.155

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• Thm 4.2: (Properties of vector addition and scalar multiplication)

Let u, v, and w be vectors in Rⁿ , and let c and d be scalars.

Elementary Linear Algebra: Section 4.1, p.156

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• Ex 5: (Vector operations in R⁴)

​

Sol: (a)

Let u=(2, – 1, 5, 0), v=(4, 3, 1, – 1), and w=(– 6, 2, 0, 3) be vectors in R⁴. Solve x for x in each of the following.

(a) x = 2u – (v + 3w)

(b) 3(x+w) = 2u – v+x

Elementary Linear Algebra: Section 4.1, p.156

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(b)

Elementary Linear Algebra: Section 4.1, p.156

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• Thm 4.3: (Properties of additive identity and additive inverse)

Let v be a vector in Rⁿ and c be a scalar. Then the following is true.

Elementary Linear Algebra: Section 4.1, p.157

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• Linear combination:

The vector x is called a linear combination of ,

if it can be expressed in the form

Elementary Linear Algebra: Section 4.1, p.158

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• Notes:

A vector in can be viewed as:

(The matrix operations of addition and scalar multiplication

give the same results as the corresponding vector operations)

or

​

a n×1 column matrix (column vector):

a 1×n row matrix (row vector):

Elementary Linear Algebra: Section 4.1, p.158

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Vector addition

Scalar multiplication

Elementary Linear Algebra: Section 4.1, p.158

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Key Learning in Section 4.1

• Represent a vector as a directed line segment.
• Perform basic vector operations in R² and represent them graphically.
• Perform basic vector operations in Rⁿ.
• Write a vector as a linear combination of other vectors.

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Keywords in Section 4.1

• ordered n-tuple：有序的n項
• n-space：n維空間
• equal：相等
• vector addition：向量加法
• scalar multiplication：純量乘法
• negative：負向量
• difference：向量差
• zero vector：零向量
• additive identity：加法單位元素
• additive inverse：加法反元素

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4.2 Vector Spaces

• Vector spaces:

Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the following axioms are satisfied for every u, v, and w in V and every scalar (real number) c and d, then V is called a vector space.

Elementary Linear Algebra: Section 4.2, p.161

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Scalar multiplication:

(6) is in V.

(7)

(8)

(9)

(10)

Elementary Linear Algebra: Section 4.2, p.161

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• Notes:

(1) A vector space consists of four entities:

V：nonempty set

c：scalar

vector addition

scalar multiplication

a set of vectors, a set of scalars, and two operations

Elementary Linear Algebra: Section 4.2, Addition

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• Examples of vector spaces:

(1) n-tuple space: Rⁿ

vector addition

scalar multiplication

Elementary Linear Algebra: Section 4.2, Addition

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(3) n-th degree polynomial space:

(the set of all real polynomials of degree n or less)

Elementary Linear Algebra: Section 4.2, Addition

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• Thm 4.4: (Properties of scalar multiplication)

Let v be any element of a vector space V, and let c be any

scalar. Then the following properties are true.

Elementary Linear Algebra: Section 4.2, p.164

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• Notes: To show that a set is not a vector space, you need

only find one axiom that is not satisfied.

Elementary Linear Algebra: Section 4.2, p.165

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• Ex 8:

V=R²=the set of all ordered pairs of real numbers

vector addition:

scalar multiplication:

Verify V is not a vector space.

Sol:

Elementary Linear Algebra: Section 4.2, p.165

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Key Learning in Section 4.2

• Define a vector space and recognize some important vector spaces.
• Show that a given set is not a vector space.

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Keywords in Section 4.2:

• vector space：向量空間
• n-space：n維空間
• matrix space：矩陣空間
• polynomial space：多項式空間
• function space：函數空間

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4.3 Subspaces of Vector Spaces

• Subspace:

: a vector space

: a nonempty subset

：a vector space (under the operations of addition and

scalar multiplication defined in V)

W is a subspace of V

Elementary Linear Algebra: Section 4.3, p.168

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• Thm 4.5: (Test for a subspace)

If W is a nonempty subset of a vector space V, then W is

a subspace of V if and only if the following conditions hold.

(1) If u and v are in W, then u+v is in W.

(2) If u is in W and c is any scalar, then cu is in W.

Elementary Linear Algebra: Section 4.3, p.168

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• Ex: Subspace of R³

• Ex: Subspace of R²

Elementary Linear Algebra: Section 4.3, p.168

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• Ex 2: (A subspace of M_2×2)

Let W be the set of all 2×2 symmetric matrices. Show that

W is a subspace of the vector space M_2×2, with the standard

operations of matrix addition and scalar multiplication.

Sol:

Elementary Linear Algebra: Section 4.3, p.169

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• Ex 3: (The set of singular matrices is not a subspace of M_2×2)

Let W be the set of singular matrices of order 2. Show that

W is not a subspace of M_2×2 with the standard operations.

Sol:

Elementary Linear Algebra: Section 4.3, p.169

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• Ex 4: (The set of first-quadrant vectors is not a subspace of R²)

Show that , with the standard

operations, is not a subspace of R².

​

Sol:

Elementary Linear Algebra: Section 4.3, p.170

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• Ex 6: (Determining subspaces of R²)

Which of the following two subsets is a subspace of R²?

(a) The set of points on the line given by x+2y=0.

(b) The set of points on the line given by x+2y=1.

Sol:

(closed under addition)

(closed under scalar multiplication)

Elementary Linear Algebra: Section 4.3, p.171

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(b)

(Note: the zero vector is not on the line)

Elementary Linear Algebra: Section 4.3, p.171

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• Ex 8: (Determining subspaces of R³)

Sol:

Elementary Linear Algebra: Section 4.3, p.172

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• Thm 4.6: (The intersection of two subspaces is a subspace)

Elementary Linear Algebra: Section 4.3, p.170

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Key Learning in Section 4.3

• Determine whether a subset W of a vector space V is a subspace of V.
• Determine subspaces of Rⁿ.

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Keywords in Section 4.3:

• subspace：子空間
• trivial subspace：顯然子空間

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4.4 Spanning Sets and Linear Independence

• Linear combination:

Elementary Linear Algebra: Section 4.4, p.175

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• Ex 2-3: (Finding a linear combination)

Sol:

Elementary Linear Algebra: Section 4.4, p.176

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Elementary Linear Algebra: Section 4.4, p.176

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Elementary Linear Algebra: Section 4.4, p.176

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If S ={v₁, v₂,…, v_k} is a set of vectors in a vector space V, then the span of S is the set of all linear combinations of the vectors in S,

• the span of a set: span (S)

• a spanning set of a vector space:

If every vector in a given vector space can be written as a linear combination of vectors in a given set S, then S is called a spanning set of the vector space.

Elementary Linear Algebra: Section 4.4, p.177

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• Notes:

Elementary Linear Algebra: Section 4.4, p.177

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• Ex 5: (A spanning set for R³)

Sol:

Elementary Linear Algebra: Section 4.4, p.177

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Elementary Linear Algebra: Section 4.4, p.177

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• Thm 4.7: (Span(S) is a subspace of V)

If S={v₁, v₂,…, v_k} is a set of vectors in a vector space V,

then

1. span (S) is a subspace of V.
2. span (S) is the smallest subspace of V that contains S.

(Every other subspace of V that contains S must contain span (S).)

Elementary Linear Algebra: Section 4.4, p.178

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• Linear Independent (L.I.) and Linear Dependent (L.D.):

: a set of vectors in a vector space V

Elementary Linear Algebra: Section 4.4, p.179

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• Notes:

Elementary Linear Algebra: Section 4.4, p.179

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• Ex 8: (Testing for linearly independent)

Sol:

Determine whether the following set of vectors in R³ is L.I. or L.D.

Elementary Linear Algebra: Section 4.4, p.180

v₁ v₂ v₃

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• Ex 9: (Testing for linearly independent)

Determine whether the following set of vectors in P₂ is L.I. or L.D.

S = {1+x – 2x² , 2+5x – x² , x+x²}

c₁v₁+c₂v₂+c₃v₃ = 0

v₁ v₂ v₃

Sol:

⇒ This system has infinitely many solutions.

(i.e., This system has nontrivial solutions.)

Elementary Linear Algebra: Section 4.4, p.181

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• Ex 10: (Testing for linearly independent)

Determine whether the following set of vectors in 2×2

matrix space is L.I. or L.D.

Sol:

c₁v₁+c₂v₂+c₃v₃ = 0

v₁ v₂ v₃

Elementary Linear Algebra: Section 4.4, p.181

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Elementary Linear Algebra: Section 4.4, p.181

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• Thm 4.8: (A property of linearly dependent sets)

A set S = {v₁,v₂,…,v_k}, k≥2, is linearly independent if and

only if at least one of the vectors v_j in S can be written as

a linear combination of the other vectors in S.

⇒ c_i ≠ 0 for some i

(⇒)

c₁v₁+c₂v₂+…+c_kv_k = 0

Pf:

Elementary Linear Algebra: Section 4.4, p.182

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Let

(nontrivial solution)

• Corollary to Theorem 4.8:

Two vectors u and v in a vector space V are linearly dependent

if and only if one is a scalar multiple of the other.

v_i = d₁v₁+…+d_i-1v_i-1+d_i+1v_i+1+…+d_kv_k

⇒ d₁v₁+…+d_i-1v_i-1-v_i+d_i+1v_i+1+…+d_kv_k = 0

⇒ c₁=d₁, …,c_i-1=d_i-1, c_i=-1,c_i+1=d_i+1,…, c_k=d_k

Elementary Linear Algebra: Section 4.4, p.182-183

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Key Learning in Section 4.4

• Write a linear combination of a set of vectors in a vector space V.
• Determine whether a set S of vectors in a vector space V is a spanning set of V.
• Determine whether a set of vectors in a vector space V is linearly independent.

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Keywords in Section 4.4:

• linear combination：線性組合
• spanning set：生成集合
• trivial solution：顯然解
• linear independent：線性獨立
• linear dependent：線性相依

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4.5 Basis and Dimension

• Basis:

V：a vector space

S spans V (i.e., span(S) = V )

S is linearly independent

Generating

Sets

Bases

Linearly

Independent

Sets

⇒ S is called a basis for V

S ={v₁, v₂, …, v_n}⊆V

Elementary Linear Algebra: Section 4.5, p.186

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(3) the standard basis for Rⁿ :

{e₁, e₂, …, e_n} e₁= (1,0,…,0), e₂= (0,1,…,0), e_n= (0,0,…,1)

Ex: R⁴

{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}

Elementary Linear Algebra: Section 4.5, p.186

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• Thm 4.9: (Uniqueness of basis representation)

If is a basis for a vector space V, then every

vector in V can be written in one and only one way as a linear

combination of vectors in S.

Elementary Linear Algebra: Section 4.5, p.188

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• Thm 4.10: (Bases and linear dependence)

If is a basis for a vector space V, then every

set containing more than n vectors in V is linearly dependent.

Pf:

S₁ = {u₁, u₂, …, u_m} , m > n

Let

u_i∈V

Elementary Linear Algebra: Section 4.5, p.189

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⇒ d_i= 0 ∀i

i.e.

Let

k₁u₁+k₂u₂+…+k_mu_m= 0

(where d_i = c_i1k₁+c_i2k₂+…+c_imk_m)

⇒ d₁v₁+d₂v₂+…+d_nv_n= 0

Thm 1.1: If the homogeneous system has fewer equations than variables, then it must have infinitely many solution.

m > n ⇒ k₁u₁+k₂u₂+…+k_mu_m = 0 has nontrivial solution

⇒ S₁ is linearly dependent

Elementary Linear Algebra: Section 4.5, p.189

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• Thm 4.11: (Number of vectors in a basis)

If a vector space V has one basis with n vectors, then every

basis for V has n vectors. (All bases for a finite-dimensional

vector space has the same number of vectors.)

Pf:

S ={v₁, v₂, …, v_n}

S'={u₁, u₂, …, u_m}

two bases for a vector space

Elementary Linear Algebra: Section 4.5, p.190

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• Finite dimensional:

A vector space V is called finite dimensional,

if it has a basis consisting of a finite number of elements.

• Infinite dimensional:

If a vector space V is not finite dimensional,

then it is called infinite dimensional.

Elementary Linear Algebra: Section 4.5, p.191

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• Notes:

(1) dim({0}) = 0 = #(Ø)

(2) dim(V) = n , S⊆V

S：a generating set ⇒ #(S) ≥ n

S：a L.I. set ⇒ #(S) ≤ n

S：a basis ⇒ #(S) = n

(3) dim(V) = n , W is a subspace of V ⇒ dim(W) ≤ n

Elementary Linear Algebra: Section 4.5, Addition

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• Ex:

(1) Vector space Rⁿ ⇒ basis {e₁ , e₂ , … , e_n}

(2) Vector space M_m×n ⇒ basis {E_ij | 1≤i≤m , 1≤j≤n}

(3) Vector space P_n(x) ⇒ basis {1, x, x², … , xⁿ}

(4) Vector space P(x) ⇒ basis {1, x, x², …}

⇒ dim(Rⁿ) = n

⇒ dim(M_m×n)=mn

⇒ dim(P_n(x)) = n+1

⇒ dim(P(x)) = ∞

Elementary Linear Algebra: Section 4.5, Addition

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• Ex 9: (Finding the dimension of a subspace)

(a) W={(d, c–d, c): c and d are real numbers}

(b) W={(2b, b, 0): b is a real number}

Sol:

(Note: Find a set of L.I. vectors that spans the subspace)

⇒ S = {(0, 1, 1) , (1, – 1, 0)}

(S is L.I. and S spans W)

⇒ S is a basis for W

⇒ dim(W) = #(S) = 2

⇒ S = {(2, 1, 0)} spans W and S is L.I.

⇒ S is a basis for W

⇒ dim(W) = #(S) = 1

Elementary Linear Algebra: Section 4.5, p.191

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• Ex 11: (Finding the dimension of a subspace)

Let W be the subspace of all symmetric matrices in M_2×2.

What is the dimension of W?

Sol:

⇒ S is a basis for W

⇒ dim(W) = #(S) = 3

Elementary Linear Algebra: Section 4.5, p.192

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• Thm 4.12: (Basis tests in an n-dimensional space)

Let V be a vector space of dimension n.

(1) If is a linearly independent set of

vectors in V, then S is a basis for V.

(2) If spans V, then S is a basis for V.

Elementary Linear Algebra: Section 4.5, p.192

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Key Learning in Section 4.5

• Recognize bases in the vector spaces Rⁿ, P_n and M_m,n
• Find the dimension of a vector space.

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Keywords in Section 4.5

• basis：基底
• dimension：維度
• finite dimension：有限維度
• infinite dimension：無限維度

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4.6 Rank of a Matrix and Systems of Linear Equations

Row vectors of A

• row vectors:

Column vectors of A

• column vectors:

|| || ||

A⁽¹⁾ A⁽²⁾ A⁽ⁿ⁾

Elementary Linear Algebra: Section 4.6, p.195

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Let A be an m×n matrix.

• Row space:

The row space of A is the subspace of Rⁿ spanned by

the row vectors of A.

Elementary Linear Algebra: Section 4.6, p.195

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• Notes:

(1) The row space of a matrix is not changed by elementary

row operations.

RS(r(A)) = RS(A) r: elementary row operations

(2) Elementary row operations can change the column space.

• Thm 4.13: (Row-equivalent matrices have the same row space)

If an m×n matrix A is row equivalent to an m×n matrix B,

then the row space of A is equal to the row space of B.

Elementary Linear Algebra: Section 4.6, p.196

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• Thm 4.14: (Basis for the row space of a matrix)

If a matrix A is row equivalent to a matrix B in row-echelon

form, then the nonzero row vectors of B form a basis for the

row space of A.

Elementary Linear Algebra: Section 4.6, p.196

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Find a basis of row space of A =

• Ex 2: ( Finding a basis for a row space)

Elementary Linear Algebra: Section 4.6, p.196

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• Notes:

a basis for RS(A) = {the nonzero row vectors of B} (Thm 4.14)

= {w₁, w₂, w₃} = {(1, 3, 1, 3), (0, 1, 1, 0), (0, 0, 0, 1)}

Elementary Linear Algebra: Section 4.6, p.196

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• Ex 3: (Finding a basis for a subspace)

Find a basis for the subspace of R³ spanned by

Sol:

a basis for span({v₁, v₂, v₃})

= a basis for RS(A)

= {the nonzero row vectors of B} (Thm 4.14)

= {w₁, w₂}

= {(1, –2, – 5) , (0, 1, 3)}

Elementary Linear Algebra: Section 4.6, p.197

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• Ex 4-5: (Finding a basis for the column space of a matrix)

Find a basis for the column space of the matrix A given in Ex 2.

Sol. (Method 1):

Elementary Linear Algebra: Section 4.6, pp.197-198

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CS(A)=RS(A^T)

(a basis for the column space of A)

a basis for CS(A)

= a basis for RS(A^T)

= {the nonzero vectors of B}

= {w₁, w₂, w₃}

• Note: This basis is not a subset of {c₁, c₂, c₃, c₄}.

Elementary Linear Algebra: Section 4.6, pp.197-198

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• Notes:

(1) This basis is a subset of {c₁, c₂, c₃, c₄}.

(2) v₃ = –2v₁+ v₂, thus c₃ = – 2c₁+ c₂ .

• Sol. (Method 2):

Leading 1 => {v₁, v₂, v₄} is a basis for CS(B)

{c₁, c₂, c₄} is a basis for CS(A)

Elementary Linear Algebra: Section 4.6, pp.197-198

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• Thm 4.16: (Solutions of a homogeneous system)

If A is an m×n matrix, then the set of all solutions of the

homogeneous system of linear equations Ax = 0 is a subspace

of Rⁿ called the nullspace of A.

Pf:

• Notes: The nullspace of A is also called the solution space of

the homogeneous system Ax = 0.

Elementary Linear Algebra: Section 4.6, p.200

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• Ex 7: (Finding the solution space of a homogeneous system)

Find the nullspace of the matrix A.

Sol: The nullspace of A is the solution space of Ax = 0.

Elementary Linear Algebra: Section 4.6, p.201

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• Thm 4.15: (Row and column space have equal dimensions)

If A is an m×n matrix, then the row space and the column

space of A have the same dimension.

dim(RS(A)) = dim(CS(A))

• Rank:

The dimension of the row (or column) space of a matrix A

is called the rank of A and is denoted by rank(A).

rank(A) = dim(RS(A)) = dim(CS(A))

Elementary Linear Algebra: Section 4.6, pp.198-199

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• Note: rank(A^T) = rank(A)

• Nullity:

The dimension of the nullspace of A is called the nullity of A.

nullity(A) = dim(NS(A))

Elementary Linear Algebra: Section 4.6, p.200

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• Notes:

(1) rank(A): The number of leading variables in the solution of Ax=0.

(The number of nonzero rows in the row-echelon form of A)

(2) nullity (A): The number of free variables in the solution of Ax = 0.

• Thm 4.17: (Dimension of the solution space)

If A is an m×n matrix of rank r, then the dimension of

the solution space of Ax = 0 is n – r. That is

n = rank(A) + nullity(A)

Elementary Linear Algebra: Section 4.6, p.202

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• Notes:

If A is an m×n matrix and rank(A) = r, then

Elementary Linear Algebra: Section 4.6, Addition

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• Ex 8: (Rank and nullity of a matrix)

Let the column vectors of the matrix A be denoted by a₁, a₂,

a₃, a₄, and a₅.

(a) Find the rank and nullity of A.

(b) Find a subset of the column vectors of A that forms a basis for

the column space of A .

(c) If possible, write the third column of A as a linear combination

of the first two columns.

Elementary Linear Algebra: Section 4.6, p.202

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Sol: Let B be the reduced row-echelon form of A.

(a) rank(A) = 3 (the number of nonzero rows in B)

Elementary Linear Algebra: Section 4.6, p.202

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(b) Leading 1

Elementary Linear Algebra: Section 4.6, p.202

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• Thm 4.18: (Solutions of a nonhomogeneous linear system)

If x_p is a particular solution of the nonhomogeneous system

Ax = b, then every solution of this system can be written in

the form x = x_p + x_h , wher x_h is a solution of the corresponding

homogeneous system Ax = 0.

Pf:

Let x be any solution of Ax = b.

Elementary Linear Algebra: Section 4.6, p.203

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• Ex 9: (Finding the solution set of a nonhomogeneous system)

Find the set of all solution vectors of the system of linear equations.

​

​

Sol:

Elementary Linear Algebra: Section 4.6, p.203

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i.e.

x_h = su₁ + tu₂ is a solution of Ax = 0

is a particular solution vector of Ax=b.

Elementary Linear Algebra: Section 4.6, p.203

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• Thm 4.19: (Solution of a system of linear equations)

The system of linear equations Ax = b is consistent if and only

if b is in the column space of A.

Pf:

Let

be the coefficient matrix, the column matrix of unknowns,

and the right-hand side, respectively, of the system Ax = b.

Elementary Linear Algebra: Section 4.6, p.204

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Then

Hence, Ax = b is consistent if and only if b is a linear combination

of the columns of A. That is, the system is consistent if and only if

b is in the subspace of R^m spanned by the columns of A.

Elementary Linear Algebra: Section 4.6, p.204

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• Ex 10: (Consistency of a system of linear equations)

Sol:

• Note:

If rank([A|b])=rank(A)

Then the system Ax=b is consistent.

Elementary Linear Algebra: Section 4.6, p.204

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(b is in the column space of A)

The system of linear equations is consistent.

• Check:

Elementary Linear Algebra: Section 4.6, p.204

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• Summary of equivalent conditions for square matrices:

If A is an n×n matrix, then the following conditions are equivalent.

Elementary Linear Algebra: Section 4.6, p.204

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Key Learning in Section 4.6

• Find a basis for the row space, a basis for the column space, and the rank of a matrix.
• Find the nullspace of a matrix.
• Find the solution of a consistent system Ax = b in the form x_p + x_h.

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Keywords in Section 4.6:

• row space : 列空間
• column space : 行空間
• null space: 零空間
• solution space : 解空間
• rank: 秩
• nullity : 核次數

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4.7 Coordinates and Change of Basis

• Coordinate representation relative to a basis

Let B = {v₁, v₂, …, v_n} be an ordered basis for a vector space V

and let x be a vector in V such that

The scalars c₁, c₂, …, c_n are called the coordinates of x relative to the basis B. The coordinate matrix (or coordinate vector) of x relative to B is the column matrix in Rⁿ whose components are the coordinates of x.

Elementary Linear Algebra: Section 4.7, p.208

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• Ex 1: (Coordinates and components in Rⁿ)

Find the coordinate matrix of x = (–2, 1, 3) in R³

relative to the standard basis� S = {(1, 0, 0), ( 0, 1, 0), (0, 0, 1)}

Sol:

Elementary Linear Algebra: Section 4.7, p.208

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• Ex 3: (Finding a coordinate matrix relative to a nonstandard basis)

Find the coordinate matrix of x = (1, 2, –1) in R³

relative to the (nonstandard) basis� B ' = {u₁, u₂, u₃}={(1, 0, 1), (0, – 1, 2), (2, 3, – 5)}

Sol:

Elementary Linear Algebra: Section 4.7, p.209

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• Change of basis problem:

You were given the coordinates of a vector relative to one

basis B and were asked to find the coordinates relative to

another basis B'.

• Ex: (Change of basis)

Consider two bases for a vector space V

Elementary Linear Algebra: Section 4.7, p.210

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Let

Elementary Linear Algebra: Section 4.7, p.210

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• Transition matrix from B' to B:

where

is called the transition matrix from B' to B

If [v]_B is the coordinate matrix of v relative to B

[v]_B‘ is the coordinate matrix of v relative to B'

Elementary Linear Algebra: Section 4.7, p.210

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• Thm 4.20: (The inverse of a transition matrix)

If P is the transition matrix from a basis B' to a basis B in Rⁿ,

then

(1) P is invertible

(2) The transition matrix from B to B' is P^–1

Elementary Linear Algebra: Section 4.7, p.210

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• Thm 4.21: (Transition matrix from B to B')

Let B={v₁, v₂, … , v_n} and B' ={u₁, u₂, … , u_n} be two bases

for Rⁿ. Then the transition matrix P^–1 from B to B' can be found

by using Gauss-Jordan elimination on the n×2n matrix

as follows.

Elementary Linear Algebra: Section 4.7, p.210

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• Ex 5: (Finding a transition matrix)

B ={(–3, 2), (4,–2)} and B' ={(–1, 2), (2,–2)} are two bases for R²

(a) Find the transition matrix from B' to B.

(b)

(c) Find the transition matrix from B to B' .

Elementary Linear Algebra: Section 4.7, p.214

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Sol:

(a)

​

​

​

​

​

​

​

B B'

I P

(the transition matrix from B' to B)

Elementary Linear Algebra: Section 4.7, p.214

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(c)

(the transition matrix from B to B')

B' B

I P^-1

Elementary Linear Algebra: Section 4.7, p.214

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• Ex 6: (Coordinate representation in P₃(x))

(a) Find the coordinate matrix of p = 3x³-2x²+4 relative to the

standard basis S = {1, x, x², x³} in P₃(x).

(b) Find the coordinate matrix of p = 3x³-2x²+4 relative to the

basis S = {1, 1+x, 1+ x², 1+ x³} in P₃(x).

Sol:

​

Elementary Linear Algebra: Section 4.7, p.215

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• Ex: (Coordinate representation in M_2x2)

Find the coordinate matrix of x = relative to

the standard basis in M_2×2.

B =

Sol:

Elementary Linear Algebra: Section 5.7, Addition

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Key Learning in Section 4.7

• Find a coordinate matrix relative to a basis in Rⁿ
• Find the transition matrix from the basis to the basis B' in Rⁿ.
• Represent coordinates in general n-dimensional spaces.

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Keywords in Section 4.7

• coordinates of x relative to B: x相對於B的座標
• coordinate matrix: 座標矩陣
• coordinate vector: 座標向量
• change of basis problem: 基底變換問題
• transition matrix from B' to B: 從 B' 到 B的轉移矩陣

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4.8 Applications of Vector Spaces

• Conic sections and rotation:

Every conic section in the xy-plane has an equation that can be written in the form

Identifying the graph of this equation is fairly simple as long as b, the coefficient of the xy-term, is zero.

When b is zero, the conic axes are parallel to the coordinate axes, and the identification is accomplished by writing the equation in standard (completed square) form.

Elementary Linear Algebra: Section 4.8, p.221

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• Standard forms of equations of conics:

Elementary Linear Algebra: Section 4.8, p.221

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Elementary Linear Algebra: Section 4.8, p.221

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Elementary Linear Algebra: Section 4.8, p.221

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• Ex 5: (Identifying Conic Sections)

The standard form of is

The axis of the parabola is vertical. Because p = −1,

the focus is the point (1, 0).

Because the focus lies below the vertex,

the parabola opens downward.

vertex at

Elementary Linear Algebra: Section 4.8, p.222

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The standard form of is

The vertices of this ellipse occur at (−5, 1) and (−1, 1), and the end points of the minor axis occur at (−3, 2) and (−3, 0).

center at

Elementary Linear Algebra: Section 4.8, p.222

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For second-degree equations that have an xy-term, the axes of the graphs of the corresponding conics are not parallel to the coordinate axes.

The required rotation θ angle (measured counterclockwise) is

standard basis

new basis

Elementary Linear Algebra: Section 4.8, p.222

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• Ex 6: (A Transition Matrix for Rotation in R²)

Sol: By Theorem 4.21

Elementary Linear Algebra: Section 4.8, p.223

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• Rotation of axes:

The general second-degree equation

can be written in the form

by rotating the coordinate axes counterclockwise through the angle θ, where θ is defined by

The coefficients of the new equation are obtained from the substitutions

and

Elementary Linear Algebra: Section 4.8, p.223

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• Ex 7: (Rotation of a Conic Section)

Perform a rotation of axes to eliminate the xy-term in

and sketch the graph of the resulting equation in the x’y’-plane.

Sol: The angle of rotation

By substituting

Elementary Linear Algebra: Section 4.8, p.224

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new basis

the vertices

use the equations

obtain

Elementary Linear Algebra: Section 4.8, p.224

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Key Learning in Section 4.8

• Use the Wronskian to test a set of solutions of a linear homogeneous differential equation for linear independence.
• Identify and sketch the graph of a conic section and perform a rotation of axes.

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Keywords in Section 4.8

• Ellipse: 橢圓
• Hyperbola: 雙曲線
• Parabola: 拋物線
• Vertex: 頂點
• Focus: 焦點

​

​

​

​

​

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• Force

Vectors have a wide variety of applications in engineering and the physical sciences. For example, to determine the amount of force required to pull an object up a ramp that has an angle of elevation θ, use the figure at the right.

In the figure, the vector labeled W represents the weight of the object, and the vector labeled F represents the required force. Using similar triangles and some trigonometry, the required force is F = W sin θ. (Verifying this.)

4.1 Linear Algebra Applied

Elementary Linear Algebra: Section 4.1, p.157

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In a mass-spring system, motion is assumed to occur in only the vertical direction. That is, the system has one degree of freedom. When the mass is pulled downward and then released, the system will oscillate. If the system is undamped, meaning that there are no forces present to slow or stop the oscillation, then the system will oscillate indefinitely. Applying Newton’s Second Law of Motion to the mass yields the second order differential equation

​

where x is the displacement at time t and ω is a fixed constant called the natural frequency of the system. The general solution of this differential equation is

​

where a₁ and a₂ are arbitrary constants. (Try verifying this.) In Exercise 41, you are asked to show that the set of all functions x(t) is a vector space.

4.2 Linear Algebra Applied

Elementary Linear Algebra: Section 4.2, p.164

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• Digital Sampling

​

Digital signal processing depends on sampling, which converts continuous signals into discrete sequences that can be used by digital devices. Traditionally, sampling is uniform and pointwise, and is obtained from a single vector space. Then, the resulting sequence is reconstructed into a continuous-domain signal. Such a process, however, can involve a significant reduction in information, which could result in a low-quality reconstructed signal. In applications such as radar, geophysics, and wireless communications, researchers have determined situations in which sampling from a union of vector subspaces can be more appropriate.

4.3 Linear Algebra Applied

Elementary Linear Algebra: Section 4.3, p.172

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• Image Morphing

​

Image morphing is the process of transforming one

image into another by generating a sequence of synthetic intermediate images. Morphing has a wide variety of applications, such as movie special effects, age progression software, and simulating wound healing and cosmetic surgery results. Morphing an image uses a process called warping, in which a piece of an image is distorted. The mathematics behind warping and morphing can include forming a linear combination of the vectors that bound a triangular piece of an image, and performing an affine transformation to form new vectors and a distorted image piece.

4.4 Linear Algebra Applied

Elementary Linear Algebra: Section 4.4, p.180

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• The RGB color model combinations of the colors red (r), green (g), and blue (b), known as the primary additive colors, to create all other colors in a system. Using the standard basis for R³, where r = (1, 0, 0), g = (0, 1, 0) and b = (0, 0, 1) any visible color can be represented as a linear combination c₁r + c₂g + c₃b of the primary additive colors. The coefficients c_i are values between 0 and a specified maximum a inclusive. When c₁ = c₂ = c₃ the color is grayscale, with c_i = 0 representing black and c_i = a representing white. The RGB color model is commonly used in computers, smart phones, televisions, and other electronic with a color display .

4.5 Linear Algebra Applied

Elementary Linear Algebra: Section 4.5, p. 190

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The U.S. Postal Service uses barcodes to represent such information as ZIP codes and delivery addresses. The ZIP + 4 barcode shown at the left starts with a long bar, has a series of short and long bars to represent each digit in the ZIP + 4 code and an additional digit for error checking, and ends with a long bar. The following is the code for the digits.

​

​

​

The error checking digit is such that when it is summed with the digits in the ZIP + 4 code, the result is a multiple of 10. (Verify this, as well as whether the ZIP + 4 code shown is coded correctly.) More sophisticated barcodes will also include error correcting digit(s). In an analogous way, matrices can be used to check for errors in transmitted messages. Information in the form of column vectors can be multiplied by an error detection matrix. When the resulting product is in the nullspace of the error detection matrix, no error in transmission exists. Otherwise, an error exists somewhere in the message. If the error detection matrix also has error correction, then the resulting matrix.

4.6 Linear Algebra Applied

Elementary Linear Algebra: Section 4.6, p. 200

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• Crystallography

​

Crystallography is the science of atomic and molecular structure. In a crystal, atoms are in a repeating pattern called a lattice. The simplest repeating unit in a lattice is a unit cell. Crystallographers can use bases and coordinate matrices in R³ to designate the locations of atoms in a unit cell. For example, the figure below shows the unit cell known as end-centered monoclinic.

​

​

​

​

One possible coordinate matrix for the top end-centered (blue) atom is

4.7 Linear Algebra Applied

Elementary Linear Algebra: Section 4.7, p. 213

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• Satellite Dish

​

A satellite dish is an antenna that is designed to transmit or receive signals of a specific type. A standard satellite dish consists of a bowl-shaped surface and a feed horn that is aimed toward the surface. The bowl-shaped surface is typically in the shape of an elliptic paraboloid. (See Section 7.4.) The cross section of the surface is typically in the shape of a rotated parabola.

4.8 Linear Algebra Applied

Elementary Linear Algebra: Section 4.8, p.223

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