Matrices #

Most of the material in this section was borrowed from this excellent series of Essence of Linear Algebra youtube videos

Linear Transformation #

linear-combination transformed-linear-combination The coordinates of the vector after a linear transformation are the same linear combination of the transformed basis vectors


matrix-basis-1 We can forma a matrix whose columns are the basis vectors

matrix-basis-2 We can calculate the coordinates of the transformed vector via the basis matrix-vector multiplication

determinant The factor by which the area after the transformation is increased.The sign of the dererminant indicates the “flipping” of the transformed space

determinant-calc How to calculate the determinant.

rank The rank of the matrix is the positive integer that indicates the number of dimensions in the output of the transformation that involves this matrix.

column-space The column space is the span of its column vectors (input space)

null-space The null space is the space of all vectors that after the transformation land on the origin

2d-to-3d 2D to 3D transformation. The column space includes two basis vectors. Three rows indicate that the basis vectors live in 3D.

Matrix Multiplication #

matrix-multiplication Matrix multiplication can be seen as two consecutive linear transformations

Inverse Matrices #


meaning-linear-system Looking for an $x$ that when it is transformed by $A$, it will land on $v$ - picture depicts multiple $x$’s not the solution that will be on top of $v$

inverse-transformation Looking at the reverse problem, we can start with $v$ and find the transformation that gives us $x$.This transformation is called the inverse of $A$: $A^{-1}$

existence-inverse Existence of inverse transformation

linear-system-solution Solution to the linear system of equations