# Solutions of Problem Set 2.1 in Introduction to linear algebra (Gilbert Strang)

This is not the formal. This is my personal collection of solutions.

ここから第5版に変わっています。海外からの閲覧が多いため主に英語です

Introduction to Linear Algebra 5th Edition (Gilbert Strang)

2.1 Vectors and Linear Equations

• row picture and column picture of `Ax = b`

### Problem Set 2.1

If you find any mistakes, please comment.

#### Row and column pictures of `Ax = b`

1
draw the planes in the row picture, and draw the vectors in the column picture.

2
Compare the equations in Problem 1 with integral multiples of them.

3
See what are changed when equation 1 is added to equation 2.

4
Find the solution with one variable fixed.

5
Find the solutions line.

6
An example three equations have no solution.

7
"singular case"

8
Combination of 4-column vectors in 4-dimensional space.

#### Multiplying matrices and vectors

9
Compute `Ax` by dot products.

10
Compute `Ax` as a combination of the columns.

11
Compute `Ax`.

12
Compute `Ax`.

13
`A(m x n) * x(n) -> b(m)`

14
Write down an equation as `Ax = b`.

#### Matrices that act in special ways on vectors

15
Identity matrix and exchange matrix.

16
90° and 180° rotation matrices.
180° rotation matrix is the same as `-I`.

17
`P: (x,y,z) -> (y,z,x)`, `Q: (y,z,x) -> (x,y,z)`
Q is inverse of P.

18
E subtracts the first component from the second component.

19
`E: (x,y,z) -> (x,y,z+x)`, `E^(-1): (x,y,z) -> (x,y,z-x)`

20
`P1: (x,y) -> (x,0)`, `P2: (x,y) -> (0,y)`

21
`R` rotates every vector through 45°.

22
dot product of two vectors.

omit 23, 24, and 25 as I cannot write MATLAB code.

#### Review the row and column pictures in 2, 3, and 4 dimensions.

26, 27
row picture and column picture.

28
row picture and column picture of four linear equations.

29
Markov matrix.

#### Challenge Problems

30
continue Markov matrix.

31
Magic matrix in which all rows and columns and diagonals add to a same number.

32
typical singular matrix.
You can see 3D interactive plots of a singular matrix example by opening the following notebook in colab.

33
If `w` is a combination of `u` and `v`, then `Aw` is the same combination of `Au` and `Av`.

34-1
Write linear equations in their matrix form.

34-2
Solve the equations.

35
Sudoku matrix