In this segment, we’ll take a look at how we compute the transformation matrix **M** that helps us navigate between multiple basis vectors. We’ll generalise the conventions on how to move from one basis to another basis so that it becomes easier while using the formula.

The fundamental equation to move between different basis vectors that we studied in the previous session is shown as follows:

Our target was to represent M in the terms of the old basis vectors and the new basis vectors.

Before we move to that part, we need to set some conventions for ease of use. These conventions are given as follows:

**B1**will represent the old basis and**v1**is the old basis representation**B2**will represent the new basis and**v2**is the new basis representation

This would mean that the equation shown just previously can be written as:

v2=M∗v1

Let’s consider this as **equation 1.**

Now, when we shift between multiple basis vectors the following equation remains valid,

B1∗v1=B2∗v2

A slight manipulation of this equation gives us the following equation:

B−12∗B1∗v1=B−12∗B2∗v2

which gets converted to:

v2=B−12∗B1∗v1

Let’s consider this as **equation 2**.

On comparing** Equation 1 and 2**, we get the following equation for M as follows:

M=B−12B1

Now let’s go ahead and solve some problems.

The following image summarises the learnings of the video above (click on the image to enlarge it)

Mainly when we’re moving between multiple basis vectors, it’s important to know that the **point’s position in space doesn’t change**. The point’s representation might be different in different basis vectors but it would be representing the same point.

Next, let’s go ahead and solve some examples in python. Download the notebook from the link below.

There are 3 major steps that need to be performed while doing **change of basis** operations in python. After you’ve set the proper conventions for moving from B1 to B2, you need to

– Store the basis vector matrices in numpy arrays. Make sure that you’re passing the rows of the basis vector matrix as arguments while creating the matrices.

– Create the transformation matrix by using the formula mentioned below:

M=B−12B1

– Finally, use the change of basis equation for finding a point’s new representation from old representation,

v2=M∗v1

**Comprehension**

Let’s say you have a point [86] in the standard basis vectors in 2-d [10] and [01]

You want to find its representation in the following basis :

[2−211]

Answer the following questions to sequentially solve the aforementioned problem.

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