ストラング線形代数イントロダクション解答

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
世界標準MIT教科書 ストラング:線形代数イントロダクション(ストラング・ギルバート)
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

備忘のためなので間違っているかもしれません。もし見つけたらご指摘いただけると幸いです。
If you find any mistakes, please comment.

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.

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.

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°.

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