Hour 19, 20 & 21 - Transformation Inverses & Coordinates

Hello everyone! Hope you all are doing well and welcome back again to the concept blog. Last time we learnt about matrices and matrix algebra. Now, we will explore properties of matrices.

Hour 19

Invertible Matrix

A matrix is said to be invertible if it has only one consistent solution. This means that when the matrix is converted in RREF, every column will have one leading one. So, the rank of the matrix will be equal to the number of columns of the matrix.

Inverse Transformation

To transform a matrix to it’s inverse, given that the matrix meets all the conditions for having an inverse, we take the identity matrix to the right hand side and then find the RREF of the matrix (which will be I too since it’s invertible). The resulting matrix on the right hand side, which was I before, will now be A inverse.

An example:

Transformation Inverses

A transformation T will take a vector in an input dimension and transform it to another vector in another output dimension. Similarly, a transformation inverse would just take the output vector and then transform it back to the input vector.

Adjugates and Formulae

The inverse of an invertible 2 x 2 matrix can be easily found by the following formula shown at the bottom picture.

Here, the determinant is ad - bc and note that any matrix multiplied to its inverse gives the identity matrix.

Hour 20, 21

Bases and Coordinates

We have always represented vectors but we never noticed that we are actually representing them relative to a certain basis - a basis which contains the standard basis vectors e1,e2,…,en.

Yes! we have been representing the vectors as a combination of these standard basis vectors all along. What if we can represent them relative to other bases too!

Coordinates for Transformations

The transformation matrices can also be represented relative to different bases.

That’s all for today! We wish you all the best for your midterms and the next blog will have some practice problems that would help to cover and brush up the concepts we have covered so far! Stay tuned!