(HOD)

Department of Mathematics

Gramin ACS college Vasantnagar, kotgyal

Tq: Mukhed Dist: Nanded

Gramin (ACS) Mahavidyalaya Vasantnagar , � Kotgyal � Dep. Of Mathematics � (B.Sc.TY)� Sem-5^th Paper no.–14^th � (Linear Algebra)� By- Dr.S.S.Zampalwad

Chapter 2: Vector spaces

Vector spaces, subspaces, basis, dimension, coordinates, row-equivalence, computations

A vector space (V,F, +, .)

• F a field
• V a set (of objects called vectors)
• Addition of vectors (commutative, associative) �
• Scalar multiplications

Examples

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• Other laws are easy to show

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• This is just written differently

• The space of functions: A a set, F a field

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• If A is finite, this is just F^|A|. Otherwise this is infinite dimensional.
• The space of polynomial functions

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• The following are different.

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Subspaces

• V a vector space of a field F. A subspace W of V is a subset W s.t. restricted operations of vector addition, scalar multiplication make W into a vector space.
• +:WxW -> W, •:FxW -> W.
• W nonempty subset of V is a vector subspace iff �for each pair of vectors a,b in W, and c in F, ca+b is in W. �(iff for all a,b in W, c, d in F, ca+db is in W.)
• Example:

• �is a vector subspace with field F.
• Solution spaces: Given an mxn matrix A

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• Example x+y+z=0 in R³. x+y+z=1 (no)
• The intersection of a collection of vector subspaces is a vector subspace
• is not.

Span(S)

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• Theorem 3. W= Span(S) is a vector subspace and is the set of all linear combinations of vectors in S.
• Proof:

• Sum of subsets S₁, S₂, …,S_k of V

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• If S_i are all subspaces of V, then the above is a subspace.
• Example: y=x+z subspace:

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• Row space of A: the span of row vectors of A.
• Column space of A: the space of column vectors of A.

Linear independence

• A subset S of V is linearly dependent if

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• A set which is not linearly dependent is called linearly independent: �The negation of the above statement

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Basis

• A basis of V is a linearly independent set of vectors in V which spans V.
• Example: Fⁿ the standard basis

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• V is finite dimensional if there is a finite basis. Dimension of V is the number of elements of a basis. (Independent of the choice of basis.)
• A proper subspace W of V has dim W < dim V. (to be proved)

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• Example: P invertible nxn matrix. P₁,…,P_n columns form a basis of F^nx1.
• Independence: x₁P₁+…+x_nP_n=0, PX=0. �Thus X=0.
• Span F^nx1: Y in F^nx1. Let X = P^-1Y. Then Y = PX. Y= x₁P₁+…+x_nP_n.
• Solution space of AX=0. Change to RX=0.

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• Basis E_j u_j =1, other u_k=0 and solve above�

• Thus the dimension is n-r:

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• Infinite dimensional example:
• V:={f| f(x) = c₀+c₁x+c₂x² + …+ c_nxⁿ}.
• Given any finite collection g₁,…,g_n there is a maximum degree k. Then any polynomial of degree larger than k can not be written as a linear combination.

• Theorem 4: V is spanned by �Then any independent set of vectors in V is finite and number is ≤ m.
• Proof: To prove, we show every set S with more than m vectors is linearly dependent. Let �be elements of S with n > m.

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• A is mxn matrix. Theorem 6, Ch 1, we can solve for x₁,x₂,…,x_n not all zero for

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• Thus

• Corollary. V is a finite d.v.s. Any two bases have the same number of elements.
• Proof: B,B’ basis. Then |B’|≤|B| and |B|≤|B’|.
• This defines dimension.
• dim Fⁿ=n. dim F^mxn=mn.
• Lemma. S a linearly independent subset of V. Suppose that b is a vector not in the span of S. Then S∪{b} is independent.
• Proof:�Then k=0. Otherwise b is in the span. �Thus,�and c_iare all zero.

• Theorem 5. If W is a subspace of V, every linearly independent subset of W is finite and is a part of a basis of W.
• W a subspace of V. dim W ≤ dim V.
• A set of linearly independent vectors can be extended to a basis.
• A nxn-matrix. Rows (respectively columns) of A are independent iff A is invertible. �(->) Rows of A are independent. Dim Rows A = n. Dim Rows r.r.e R of A =n. R is I -> A is inv.�(<-) A=B.R. for r.r.e form R. B is inv. AB^-1 is inv. R is inv. R=I. Rows of R are independent. Dim Span R = n. Dim Span A = n. Rows of A are independent.

• Theorem 6. � dim (W₁+W₂) = dim W₁+dimW₂-dimW₁∩W₂.
• Proof:
• W₁∩W₂ has basis a₁,…,a_k. W₁ has basis a₁,..,a_k,b₁,…,b_m. W₂ has basis a₁,..,a_k,c₁,…,c_n.
• W₁+W₂ is spanned by a₁,..,a_k,b₁,…,b_m ,c₁,…,c_n.
• There are also independent.
• Suppose

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• Then

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• By independence z_k=0. x_i=0,y_j=0 also.

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Coordinates

• Given a vector in a vector space, how does one name it? Think of charting earth.
• If we are given Fⁿ, this is easy? What about others?
• We use ordered basis:�One can write any vector uniquely

• Thus,we name

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Coordinate (nx1)-matrix (n-tuple) of a vector.� For standard basis in Fⁿ, coordinate and vector are the same.

• This sets up a one-to-one correspondence between V and Fⁿ.
• Given a vector, there is unique n-tuple of coordinates.
• Given an n-tuple of coordinates, there is a unique vector with that coordinates.
• These are verified by the properties of the notion of bases. (See page 50)

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Coordinate change?

• If we choose different basis, what happens to the coordinates?
• Given two bases
• Write

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• X=0 iff X’=0 Theorem 7,Ch1, P is invertible
• Thus, X = PX’, X’=P^-1X.

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• Example {(1,0),(0,1)}, {(1,i), (i,1)}
• (1,i) = (1,0)+i(0,1)�(i,1) = i(1,0)+(0,1)
• (a,b)=a(1,0)+b(1,0): (a,b)_B =(a,b)
• (a,b)_B’ = P^-1(a,b) = ((a-ib)/2,(-ia+b)/2).
• We check that (a-ib)/2x(1,i)+�(-ia+b)/2x(i,1)=(a,b).