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# Linear Transformation

For 1 โ 4, we will conder the linear transformation P that projects vectors in ๐2 onto the line ๐ฆ = ๐ฅ. The

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perpendicularly to the line ๐ฆ = ๐ฅ. The โshadowโ that any vector casts on that line is the projection of

that vector. Here are some examples:

To find the matrix of this linear transformation (with respect to the standard basis) we just need to

figure out what this linear transformation does to the two standard basis vectors [ 1 0 ] and [0 1 ].

ย Letโs focus on the former:

Looking at that right triangle formed by the origin, the vector e1 and the projection, we can deduce that

the angles are as labeled. The hypothenuse is length 1, and the other two sides are equal, so squaring

the sides, adding them and setting them equal to 1 tells us that each side is length โ2

2 . So the projection

vector must be the vector [

โ2

2 cos

๐

4

โ2

2 sin

๐

4

] = [

1

2 1

2

].

A similar calculation (or staring at the picture convinced you that itโs symmetric) tells us that the

projection of e2 is also [12 12].

So the matrix of this linear transformation (with respect to the standard basis) is ๐ด = [ 1 2โ

1 2โ

1 2โ

1 2โ ]. Thatโs

the main thing we need.

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