Hour 26, 28 & 29 - Orthogonality, Transpose & Determinants

Hey everyone! Welcome to the final blog of this semester. Final exams are knocking at the door and we are sorry that we could not fully cover all the topics in our blog. But the blog will continue to be updated so that future students taking Math 133 can be benefitted too!

Hour 26

Transpose

The transpose of a matrix is just another matrix with the rows and columns switched with the original matrix.

If a matrix has dimensions m x n, then its transpose will have dimensions n x m.

Orthogonality

We have seen and drawn many vectors on graphs and two vectors have a certain angle between them. If two vectors are 90 degrees to each other, they are said to be perpendicular/orthogonal to each other. This means that the dot product of the two vectors must be 0.

Orthogonal Complement of a Subspace

Not only vectors, subspaces can also have other subspaces which are perpendicular to each other. For example, a plane can be perpendicular to another plane.

Some Important Relations Involving Transpose

Here are some very important relations between matrices and transpose:

Hour 28 & 29

Determinants

A determinant is an important property for any n x n matrix. The determinant of the matrix is a constant value and it can determine a lot about the properties of the matrix.

Finding Determinant (2x2 Matrix)

To find the determinant of an nxn matrix, there is an algorithm for it. But first, let us look at the method to find the determinant of a 2x2 matrix. The determinant of a 2x2 matrix can be calculated in the following way:

Finding Determinant(nxn Matrix)

The method shown below can be used to find the determinant of any nxn matrix. This is shown on a 3x3 matrix as an example.

The answer would also be the same if the expansion of cofactors was done along any column too!

Determinants and Area

It is very interesting that the determinant again comes up when calculating the area of a parallelogram spanned by two vectors in R2. The area of the parallelogram turns out to be the determinant of the matrix formed when the columns of matrix are the two vectors.

You guys can also find more interesting properties about determinants of matrices and you can experiment by changing rows and columns, multiplying/adding/subtracting constants or rows with a matrix to see what effect it has on the determinant of the matrix.

That’s all for today and this is the last blog for this Math 133 Fall 2021 semester. We hope everyone of you enjoyed the course and best of luck for the finals!