Matrix operations have always been hard to understand; I still need to start learning from the basics of mathematics.
Inverse matrices, Column Space, Rank, Null space
Matrix operations are used to solve linearly related equations. Given the matrix of a linear transformation and the transformed vector, performing inverse matrix operations can find the original vector.
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The geometric meaning of rank is the dimension of the space after the linear transformation. Rank 2 means the transformed space is two-dimensional. For a 2x2 matrix, the transformation can result in a 2D, 1D, or 0D space.
The set of these transformed spaces is called the column space of the matrix, that is, the space spanned by the columns (basis vectors) of the matrix.
If the rank reaches the maximum, equal to the number of columns (column space), it is full rank.
For example, a 3x3 matrix transforming to still be a three-dimensional space has rank 3, which is full rank.
How to compute the inverse matrix
P9 Non-Square Matrix
A 3x2 matrix physically represents mapping a vector from two-dimensional space to three-dimensional space, where only the basis vectors (\hat{i}) and (\hat{j}) exist. A 2x3 matrix maps a three-dimensional vector to two dimensions.
P10 Dot Product and Duality
The dot product of identical-looking matrices (such as two 2D matrices) physically represents the length of the projection of one vector onto another multiplied by the length of the other vector
Don’t understand
P12 Section 08 Part 2 - Viewing the Cross Product from the Perspective of Linear Transformations
I gave up spending a whole day to understand the cross product
But I can understand two lines of it!
P13 Basis Transformation
OK
P14 Eigenvectors and Eigenvalues
On the first day, I still don’t understand it; I will review it again tomorrow.