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84grandmarquis December 1st, 2013 12:18 PM

Finding the Matrix P for the standard basis and basis B
 
Consider the orthogonal basis B= {u1= [/latex]2\\1\[/latex] and u2= [/latex]-1\\2\[/latex] and let P be the projection onto u1

Find the matrix for P in the standard basis

and

Find the matrix for P in the basis B

I know how u1 and and u2 look, and how they form a right angle at the origin and are in the 1st and second quadrants respectively but other than that I'm confused

sorry this is a repeat, please refer to my other post

HallsofIvy December 3rd, 2013 09:18 AM

Re: Finding the Matrix P for the standard basis and basis B
 
Let a vector be <x, y>. Since <2, 1> and <1, 2> are independent, we can write <x, y> as a linear combination of them. That is, there are numbers, a and b, such that a<2, 1>+ b<1, 2>= <2a+ b, a+ 2b>= <x ,y>. That is the same as saying that 2a+ b= x and a+ 2b= y. Solve those equations for a and b in terms of x and y. a<2, 1>, for that a, will be the projection of <x, y> on <2, 1>.

Now, do you know how to write a linear transformation as a matrix in a given ordered basis? Apply the matrix to each basis vector in turn, writing each result as a linear combination of the basis vectors. The coefficients form the columns in the matrix.


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