1. Orthogonal Projections
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Orthogonal Projection
For a vector u in ℝn we would like to decompose any other vector in ℝn into the sum of two vectors, one a multiple of u, and the other orthogonal to u.
That is, we wish to write:
𝑦 = 𝑦 + 𝑧
Where
𝑦 =𝛼 𝑢
for some scalar α, and z is a vector orthogonal to u.
Here is a formula. To get this, just set u•z=0 and rearrange.
𝑦 = 𝑦 ∙ 𝑢 𝑢 ∙ 𝑢 𝑢
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Orthogonal Projection
For a vector u in ℝn we would like to decompose any other vector in ℝn into the sum of two vectors, one a multiple of u, and the other orthogonal to u.
That is, we wish to write:
𝑦 = 𝑦 + 𝑧
Where
𝑦 =𝛼 𝑢
for some scalar α, and z is a vector orthogonal to u.
Another version of the formula. This one shows the unit vectors in the direction of u.
𝑦 = 𝑦 ∙ 𝑢 𝑢 𝑢 𝑢
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Orthogonal Projection
Here is an example using vectors in ℝ2.
𝑦 = 4 7
𝑢 = 2 1
Find the orthogonal projection of y onto u.
𝑦 = 𝑦 ∙ 𝑢 𝑢 ∙ 𝑢 𝑢
𝑦 = 𝑢 = 6 3 𝑦
Subtracting yields the vector z, which is orthogonal to u. Check this by finding that z•u=0. 𝑧 𝑢
𝑧 = 𝑦 − 𝑦 = −2 4
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Orthogonal Projection
One important consequence of the previous calculation is that we have manufactured an orthogonal basis for the vector space ℝ2. This idea can be very useful in a variety of situations.
The original set of vectors was a basis for ℝ2, but the vectors were not orthogonal. To find an orthogonal basis, we simply found the projection of one vector onto the other, then subtracted it, leaving an orthogonal vector.
We can go a step further and find an orthonormal basis by simply dividing each vector by its magnitude.
Original basis
Orthogonal basis
Orthonormal basis
In this example, we only needed two basis vectors (ℝ2 is 2-dimensional), but if we are dealing with a larger space this process can be repeated to find as many vectors as necessary.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Orthogonal Projection
The process of constructing an orthonormal basis in the way we have described is called the Gram-Schmidt process.
Here is another example, this time with vectors in ℝ3. The given set of vectors form a basis for ℝ3. Use the Gram-Schmidt process to find an orthonormal basis for ℝ3.
Step 1 is to find the projection of v2 onto v1, and subtract it from v2, leaving a new vector that is orthogonal to v1. I will call this new vector v2*.
Can scale vector to avoid fractions
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Orthogonal Projection
We now have two orthogonal vectors. To manufacture the third one we will project v3 onto both of these, and subtract those projections to obtain a vector that is orthogonal to both v1 and v2*. Call this one v3*.
We can scale these vectors however we want, so for convenience we can use the following orthogonal set:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Orthogonal Projection
The last step is to normalize the vectors. Simply find the length of each vector and divide, obtaining a vector of length 1 that points the same direction.
Here is an orthonormal set of vectors that spans ℝ3.
If we make a matrix with these vectors as columns we get a very convenient orthogonal (orthonormal) matrix with the property that UT=U-1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB