This is not the formal. This is my personal collection of solutions.
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
draw the planes in the row picture, and draw the vectors in the column picture.
Compare the equations in Problem 1 with integral multiples of them.
See what are changed when equation 1 is added to equation 2.
Find the solution with one variable fixed.
Find the solutions line.
An example three equations have no solution.
Combination of 4-column vectors in 4-dimensional space.
Multiplying matrices and vectors
Ax by dot products.
Ax as a combination of the columns.
A(m x n) * x(n) -> b(m)
Write down an equation as
Ax = b.
Matrices that act in special ways on vectors
Identity matrix and exchange matrix.
90° and 180° rotation matrices.
180° rotation matrix is the same as
P: (x,y,z) -> (y,z,x),
Q: (y,z,x) -> (x,y,z)
Q is inverse of P.
E subtracts the first component from the second component.
E: (x,y,z) -> (x,y,z+x),
E^(-1): (x,y,z) -> (x,y,z-x)
P1: (x,y) -> (x,0),
P2: (x,y) -> (0,y)
R rotates every vector through 45°.
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.
row picture and column picture.
row picture and column picture of four linear equations.
continue Markov matrix.
Magic matrix in which all rows and columns and diagonals add to a same number.
typical singular matrix.
You can see 3D interactive plots of a singular matrix example by opening the following notebook in colab.
w is a combination of
Aw is the same combination of
Write linear equations in their matrix form.
Solve the equations.