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

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For a vector u in ℝ n we would like to decompose any other vector in ℝ n into the sum of two vectors, one a multiple of u, and the other orthogonal to u. Where Orthogonal Projection That is, we wish to write: for some scalar α, and z is a vector orthogonal to u.

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

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For a vector u in ℝ n we would like to decompose any other vector in ℝ n into the sum of two vectors, one a multiple of u, and the other orthogonal to u. Where Orthogonal Projection That is, we wish to write: 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.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Here is an example using vectors in ℝ 2. Orthogonal Projection Find the orthogonal projection of y onto u.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Here is an example using vectors in ℝ 2. Orthogonal Projection Find the orthogonal projection of y onto u. Subtracting yields the vector z, which is orthogonal to u. Check this by finding that zu=0.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Here is an example using vectors in ℝ 2. Orthogonal Projection Find the orthogonal projection of y onto u. Subtracting yields the vector z, which is orthogonal to u. Check this by finding that zu=0.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB One important consequence of the previous calculation is that we have manufactured an orthogonal basis for the vector space ℝ 2. This idea can be very useful in a variety of situations. Orthogonal Projection The original set of vectors was a basis for ℝ 2, 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.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB One important consequence of the previous calculation is that we have manufactured an orthogonal basis for the vector space ℝ 2. This idea can be very useful in a variety of situations. Orthogonal Projection The original set of vectors was a basis for ℝ 2, 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. Original basisOrthogonal basisOrthonormal basis We can go a step further and find an orthonormal basis by simply dividing each vector by its magnitude.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB One important consequence of the previous calculation is that we have manufactured an orthogonal basis for the vector space ℝ 2. This idea can be very useful in a variety of situations. Orthogonal Projection The original set of vectors was a basis for ℝ 2, 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. Original basisOrthogonal basis In this example, we only needed two basis vectors ( ℝ 2 is 2-dimensional), but if we are dealing with a larger space this process can be repeated to find as many vectors as necessary. Orthonormal basis We can go a step further and find an orthonormal basis by simply dividing each vector by its magnitude.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 ℝ 3. The given set of vectors form a basis for ℝ 3. Use the Gram-Schmidt process to find an orthonormal basis for ℝ 3. Orthogonal Projection

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 ℝ 3. The given set of vectors form a basis for ℝ 3. Use the Gram-Schmidt process to find an orthonormal basis for ℝ 3. Orthogonal Projection Step 1 is to find the projection of v 2 onto v 1, and subtract it from v 2, leaving a new vector that is orthogonal to v 1. I will call this new vector v 2 *.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We now have two orthogonal vectors. To manufacture the third one we will project v 3 onto both of these, and subtract those projections to obtain a vector that is orthogonal to both v 1 and v 2 *. Call this one v 3 *. Orthogonal Projection

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We now have two orthogonal vectors. To manufacture the third one we will project v 3 onto both of these, and subtract those projections to obtain a vector that is orthogonal to both v 1 and v 2 *. Call this one v 3 *. Orthogonal Projection 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 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. Orthogonal Projection Here is an orthonormal set of vectors that spans ℝ 3.