How do I find the matrix representation for the orthogonal projection onto vector v?

Say I have a vector [1 3]^t. How do I find the matrix representation for Pv by finding what it does to the standard basis vectors, e1 and e2?|||I'm not sure what you are asking. If you want to project a vector V onto a standard basis composed of E1 and E2, then:





If E1 and E2 are orthonormal, then any vector V = (V.E1)E1 + (V.E2)E2.





Notes:


- E1 and E2 are orthonormal if they are perpendicular and have length 1


http://en.wikipedia.org/wiki/Orthonormal鈥?/a>





- (A.B) means the dot product of the vectors A and B. The results is a scalar that gives the length of component of B parallel to A times the length of A (which is equal to the length of the component of A parallel to B times the length of B)


http://en.wikipedia.org/wiki/Dot_product





If, on the other hand, you have some linear transformation, then you can find its matrix representation by seeing what it does to E1 and E2.





Let the transform be T:


T11 T12


T21 T22





Then since E1 is (by definition) %26lt;1, 0%26gt;, if T(E1) = %26lt;X1, Y1%26gt;, then:


T11 = X1 and T21 = Y1





Similarly, since E2 = %26lt;0, 1%26gt;, if T(E2) = %26lt;X2, Y2%26gt; you get:


T12 = X2 and T22 = Y2