In the previous segment, you saw a demonstration on how the change of basis led to dimensionality reduction. Let’s go ahead and understand the elegant way of doing the same calculations.

[**Important** **Note** – There’s an error in the video below. The **ft** and **cm** corresponding to the New Basis Representation and the Old Basis Representation have been interchanged. The correct one is shown below]

As you saw above, when you have one dimension, the calculations for the change of basis are pretty straightforward. All you need to do here is to multiply the factor M which gives you the method of transforming from one basis to another.

But when the transformation requires 2 or more dimensions, what to do then? Let’s find out.

You saw that when more than one dimensions are involved, M becomes a matrix rather than a simple scalar. In this case, the first equation remains the same, just that here M is a matrix instead of a scalar.

Note that in the above example, in the old basis, representation of a data point is [5.4121.3] and you have to convert it to the new basis. Hence, the M here will be the old basis representation which is [30.48000.45].

But what if you want to go the other way around? Surely, you can’t go ahead and simply take a reciprocal right? This is where the concept of **matrix inverse** comes into the picture.

As you saw in the demonstration above the original matrix gets inversed when we want to go the other way around. Therefore, the equation remains the same in both the cases, but here the M−1 would mean the inverse of the matrix rather than a simple reciprocal.

So to summarise what you saw in the video, M=[30.48000.454]which is the matrix that shows the change of basis from ft/lbs to cm/kgs

and M−1=[0.0328002.205] which shows the changes from cm/kgs to ft/lbs

In the next segment, we’ll go ahead and generalise the idea of the change of basis using some more solved examples.

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