期中考總複習

常見題型

期中考總複習

常見題型: Solving System

of Linear Equation

送分題

(2013)

(2014)

(2018)

Solving System of �Linear Equation

(2015)

反推 RREF

RREF([A b]) =

4 x 6

Solving System of �Linear Equation

(2015)

反推 RREF

RREF([A b]) =

4 x 6

free

free

3 basic

Solving System of �Linear Equation

(2015)

反推 RREF

RREF([A b]) =

Solve Ax=0

Solving System of �Linear Equation

(2015)

RREF([A b])

[A’ b’]

RREF([A’ b’])

期中考總複習

常見題型: Determinant

Determinant

(2018)

(2012)

Determinant

(2015)

https://drive.google.com/file/d/1FCz1oB46ulzJx9ugdXiFJR_90J5ROnBM/view?usp=sharing (P8 – P11)

det(AB) = det(A) det(B)

PA = R

https://drive.google.com/file/d/1TfKUor-7wfcewFtjrmFmbFlX9avBMeZp/view?usp=sharing (P7)

P = E_kE_k-1……E₂E₁

det(R) = det(E_k) det(E_k-1) … det(E₂) det(E₁) det(A)

The answer is at

https://drive.google.com/file/d/1FCz1oB46ulzJx9ugdXiFJR_90J5ROnBM/view?usp=sharing (P4)

E: elementary matrix

期中考總複習

常見題型: Linear Function

Linear Function

• Determining a function is linear or not.
• Determining a function is onto or one-to-one.

one-to-one

onto

A: mxn

n

n

m

m

The columns of A are independent.

Linear Function

(2014)

Function

(2015)

期中考總複習

常見題型: Matrix Inverse

Invertible

A must be one-to-one

A must be onto

Matrix Inverse

(2018)

(2016)

(2016)

會不會算 A^-1

Inverse 的各種性質

Inverse v.s. Row operation

Matrix Inverse

• 會不會算 A^-1
• https://drive.google.com/file/d/1DhsYYhM7jmJk9gE8OfY8m8KuVpeXZ74n/view?usp=sharing
• https://youtu.be/YnvAIsjYBFg
• Inverse 的各種性質
• https://drive.google.com/file/d/19xRjv261kE8WPwr2aAvdxaJiKbx9pLxd/view?usp=sharing
• https://youtu.be/KnF2KvZa9tU
• Inverse v.s. Row operation
• https://drive.google.com/file/d/1TfKUor-7wfcewFtjrmFmbFlX9avBMeZp/view?usp=sharing
• https://youtu.be/BVyWqkkEUKA

​

期中考總複習

常見題型: Subspace

Subspace

• 一定要會: 檢查一個 vector set 是不是 subspace

(2016)

注意:這不是聯集!

……

1

2

3

期中考總複習

常見題型: Dependent & Independent

Dependent/Independent

(2015)

Dependent/Independent

Q

……

……

Dependent

Independent

Invertible

Dependent

Independent

(2015)

(2015)

期中考總複習

常見題型:

Rank v.s. Matrix Multiplication

Rank A (revisit)

= Dim (Row A)

= Dim (Col A^T)

Full Rank: Rank = n & Rank = m

• The size of A is mxn

Rank A = n

The columns of A are linearly independent.

Ax = b has at most one solution

A is square or 高瘦

RREF of A:

All columns are pivot columns.

1

0

0

0

1

0

0

0

Full Rank: Rank = n & Rank = m

• The size of A is mxn

Rank A = m

Ax = b always have solution (at least one solution) for every b in R^m.

The columns of A generate R^m.

Every row of R contains a pivot position (leading entry).

A is square or 矮胖

1

0

1

0

1

1

0

0

0

0

0

0

Rank

Col A = Col AB

(2012)

Rank

Rank A = Rank A^T

det A = det A^T

Col A =

https://drive.google.com/file/d/1brJ2BzNzxix2rozTOx7EAYY_AVJdto9g/view?usp=sharing (P4)

Rank – Intuitive Explanation

• If A is a m x n matrix, and B is a n x k matrix.

​

​

​

​

B

A

Rank – Intuitive Explanation

• If A is a m x n matrix, and B is a n x k matrix.

​

​

​

​

B

A

=

=

Rank B = n

期中考總複習

常見題型: Dimension & Basis

Review

• Col A (range of A), dim(Col A) = Rank(A)
• Basis: Finding pivot columns of A
• Null A, dim(Null A) = Nullity(A)
• Basis: Solving Ax = 0
• Row A = Col A^T, dim(Row A) = dim (Col A^T) = Rank(A^T)
• Basis: Finding pivot columns of A^T

​

​

​

​

Original Matrix A v.s. its RREF R

= Rank A

Three Associated Subspaces

• A is an m x n matrix

Basis?

Dimension?

Col A

Null A

Row A

in R^m

in Rⁿ

in Rⁿ

= Col A^T

A

A

Zero vector

range

Summary

Col A

Null A

Row A

Rank A

Nullity A

Rank A

= n - Rank A

A is an m x n matrix

Dimension

Basis

The pivot columns of A

The vectors in the parametric representation of the solution of Ax=0

The nonzero rows of the RREF of A

Dimension & Basis

(2015)

(2016)

=

(2019)