Inverse matrices, column space and null space

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Usefulness of matrices:

• used in computer graphics, and robotics, for manipulation of space.
• solve a system of equations - a list of unknown variables and a list of equations relating to them.

Linear system of equations:

2x+5y+3x=-3
4x+0y+8z=0
1x+3y+0z=2

This is a sample for system of equations, we use 0, when there is a variable missing in the equation.

$$\begin{array}{rl}2x+5y+3z& =-3\\ 4x+0y+8z& =0\\ 1x+3y+0z& =2\end{array}\Rightarrow \left[\begin{array}{ccc}2& 5& 3\\ 4& 0& 8\\ 1& 3& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-3\\ 0\\ 2\end{array}\right]$$

This can be written as A$\vec{x}$ = $\vec{v}$

This can be thought of as, we have a vector $\vec{x}$ which after applying the transformation A, turns into vector $\vec{v}$

The solution is different for 2 cases:

• if the Transformation squishes the space into a lower dimension, i.e, det(A) = 0
• if it leaves the space as it is, without reducing, i.e, det(A) != 0
  • only one vector $\vec{x}$ that will land on $\vec{v}$

Inverse matrices:

• The inverse of a Transformation A is called $A^{-1}$, Applying the 2 transformation one after another will leave the space unchanged.
• if A is clockwise rotation, then $A^{-1}$ is anti-clockwise rotation to the same degree.
• if det(A) = 0, then there does not exist an inverse matrix

Identity Transformation

A transformation that leaves the space unchanged, i.e, the $\hat{i}$ and $\hat{j}$ remain where they started.

Rank of a Transformation

• The dimension of the output of a Transformation is its Rank
• if a transformation changes a 3D space into a line, then its Rank is 1

Column Space

• set of all possible vectors that can be created in the space after a matrix transformation is Column Space.
• or the span of the columns of the matrix, the columns are considered as the basis vectors.

Rank of a Transformation is the number of dimensions in the column space

Null space or Kernel:

• All the vectors that land on the origin after the transformation.

Questions:

• if a det(A) is 0 then how many dimensions is reduced from the original vector?
  • The determinant will be 0 irrespective of how many dimensions is reduced.