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December 1st, 2013, 12:19 PM   #1
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Finding the Matrix P for the standard basis and basis B

Consider the orthogonal basis B= {u1= and u2= 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
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December 1st, 2013, 12:44 PM   #2
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Re: Finding the Matrix P for the standard basis and basis B

Quote:
Originally Posted by 84grandmarquis
Consider the orthogonal basis B= {u1= and u2= } 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
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December 2nd, 2013, 09:11 AM   #3
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Re: Finding the Matrix P for the standard basis and basis B

Quote:
Originally Posted by 84grandmarquis
Consider the orthogonal basis B= {u1= and u2= 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
Let <x, y> be any vector in . Then it can be written as for some numbers a and b. They must, of course, solve 2a+ b= x and a+ 2b= y. Solve those for a and b in terms of x and y. The "projection of <x, y> onto u1" is au1= a<2, 1>.

Do you know how to find the matrix representation of a linear transformation in a given basis? Apply the linear transformation to each basis vector, in turn, and write the result as a linear combination of those basis vectors. The coefficients of each vector form the columns of the matrix.
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