Hour 17 & 18 - Matrix Algebra & Compositions

Hey everyone and welcome to our blog again! Hope everyone’s having fun this Halloween! Today we will learn about the operations we can do with matrices and compostions of linear transformations!

Hour 17

Matrix Algebra

We all know what matrices are and they can be added, subtracted and even multiplied! However, there are certain conditions that must be met before you can perform algebraic operations on matrices.

Adding Matrices

Two matrices can be added with each other. The condition to be satisfied is that both the matrices must have the same dimensions. Each element of one matrix is added to the element in the same position of the other matrix. The result must be a matrix containing the same dimensions.

The same is true for subtracting matrices too! Just replace the + with a - sign.

Scalar Multiplication

A matrix can be multiplied with a scalar value, which can be any real number. The result will be a matrix of the same dimensions but each element will be multiplied by that scalar too.

Matrix Multiplication

Two matrices can be multiplied with one another and there are conditions that must hold true if two matrices can be multiplied. For two matrices A and B, AB is only valid when the number of columns of A equals the number of rows of B. The resulting matrix will have rows equal to the number of rows of A and columns equal to the number of columns of B.

Note that AB is not the same as BA. Also, the following properties hold:

1) A(B+C)= AB+AC

2) k(AB) = (kA)B = A(kB)

3) (AB)C = A(BC)

Hour 18

Compositions

Last week we learnt about what linear transformations are. Transformations can also occur multiple times. An already transformed vector can be further transformed by a different transformation. If T and S are linear transformations and a vector x is transformed first by S and then by T, the result will be T(S(x)). This is denoted as T o S.

What’s the Transformation Matrix for the Composition?

We know that linear transformations have transformation matrices. Now, since the composition is also a linear transformation, it must also have a transformation matrix itself!

Conditions Required for Compositions

Since compositions involve multiple linear transformations, there are certain conditions that needs to be satisfied by these linear transformations for them to actually be a composition.

So, if AB is multipliable, the composition can be made and product AB is the transformation matrix for that composition. In the photo above, note that we multiply B with the vector first and then multiply A with the result.

That’s all for today! See you in the next blog and Happy Halloween!