Hour 9,10 & 11 - Linear Transformations

Hour 9

Hey everyone! Welcome to our blog once again! Today we will learn about the concepts of linear transformations.

Linear Transformations

We have always seen how functions map one value(input) to another value(output). We can apply the same concept and find out that vectors can also be transformed to different vectors, even belonging to different dimensions too!

As we can see, the 4x1 input vector in R4 got transformed into a 2x1 vector in R2 and the transformation matrix is the matrix A.

Properties

Not all transformations are linear. Linear transformations have specefic criteria that needs to be fulfilled.

Preserves Addition

T is said to preserve vector addition if T(x+y) = T(x) + T(y) for all x and y in input dimension.

Preserves Scalar Multiplication

T is said to preserve scalar multiplication if T(kx) = kT(x) for all x in input dimension and scalar k.

Transformation Matrix A

We have defined T to be a linear transformation but what is the matrix representation of T? Every linear transformation T can be represented by a matrix A such that Ax = T(x).

The transformations of the basis vectors by T are the columns of the transformation matrix A. This means that the kth column of A is T(ek) where ek is the kth standard basis vector.

Hour 10

Geometric Representation of Linear Transformations

The standard basis vectors are really useful in visualising what the transformation T is actually doing.

Here A is the transformation matrix for T which represents a reflection across the y axis.

PROJECTIONS

The word projection means to cast an object onto something. Similarly vectors can also be cast onto lines and planes!

It can be helpful to visualise it such that the projection of x on line L is the shadow of x on the line if the sun was up perpendicular to L.

The properties that the projection of a vector already in the projection of x is the vector itself and the projection of a vector perpendicular to projection of x is 0 vector is crucial in finding the transformation matrix A because we can find the transformation of the basis vectors eventually as those special vectors can be written in terms of the basis vectors.

Hour 11

Projections on a Plane

So far we have seen the projection of a vector on a line. Vectors can also be projected onto a plane.

Any vector on the plane and vectors perpendicular to the plane are very useful because we always know their projections. This can be used to find the transformation matrix for the projection, just like for projections on lines.

Some helpful links

Tutorial -> https://www.youtube.com/watch?v=kYB8IZa5AuE

That’s all for today folks! Stay tuned for the next blog :’)