From the course: Machine Learning Foundations: Linear Algebra

Matrices changing basis

From the course: Machine Learning Foundations: Linear Algebra

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Matrices changing basis

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- [Instructor] Sometimes, we want to express vectors in a different basis. It is useful in many types of matrix computations. A change of basis matrix is a matrix that translates vector representations from one basis, such as the standard coordinate system, to another basis. It allows us to perform transforms in the case when the new basis vectors are not orthogonal to each other. Let's see what it means in a simple example. Our coordinate system consists of the basis vectors e1 that is equal to 1,0 and e2 that is equal to 0,1. Imagine we define an alternative vector space, with basis vectors f1 that is equal to 1,3 and f2 that is equal to 2,1, and we have a vector a, 1,1, that is represented in coordinates of that vector space. When we construct the matrix of the new basis vectors f1 and f2 and multiply it with our vector 1,1, we get the vector 3,4 as a result. We will call a matrix constructed of new basis vectors f1 and f2 transformation matrix A. It represents the change of basis from the alternative vector space to standard vector space. Now, you may wonder how to do reverse transformation from alternative vector space to the standard coordinate system. To achieve this, we have to find the inverse of the transformation matrix. For a two-by-two matrix A, there is a simple formula to calculate A-inverted. When we plug in the numbers, we get -1/5 multiplied with the matrix that has elements 1, -2, -3, and 1. And when we do scalar matrix multiplication, we get our new transformational matrix, A-inverted. We can check our result by multiplying matrix A with matrix A-inverted. And as you can see, we get the identity matrix, as expected.

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