# Matrices #

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

## Linear Transformation #

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

*We can forma a matrix whose columns are the basis vectors*

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

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

*How to calculate the determinant.*

*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.*

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

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

*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 can be seen as two consecutive linear transformations*

# Inverse Matrices #

*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 $v$*

*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 of inverse transformation*

*Solution to the linear system of equations*