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.
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.
6
An example three equations have no solution.
8
Combination of 4-column vectors in 4-dimensional space.
Multiplying matrices and vectors
10
Compute Ax
as a combination of the columns.
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.
Challenge Problems
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.