# Find the standard matrix of an orthogonal projection

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• Apr 22nd 2010, 11:27 PM
eiad77
Find the standard matrix of an orthogonal projection
Problem 4. Let W = Span[w1; w2], where
w1 = [3 0 4]
w2 = [4 5 -3]

If T (from R^3 to R^3) is the orthogonal projection onto W, find the standard matrix of T.

I did this by finding the orthogonal projections of e1, e2, and e3 on span W and constructing the transformation from [e1 e2 e3]

My answer was :
0.680 0.400 0.240
0.400 0.500 -0.300
0.240 -0.300 0.820

Is this correct? I double checked by finding that the applying the transformation matrix to w1 and w2 returned w1 and w2 which is what it should do. Thanks!
• Apr 24th 2010, 05:21 AM
Opalg
Quote:

Originally Posted by eiad77
Problem 4. Let W = Span[w1; w2], where
w1 = [3 0 4]
w2 = [4 5 -3]

If T (from R^3 to R^3) is the orthogonal projection onto W, find the standard matrix of T.

I did this by finding the orthogonal projections of e1, e2, and e3 on span W and constructing the transformation from [e1 e2 e3]

My answer was :
0.680 0.400 0.240
0.400 0.500 -0.300
0.240 -0.300 0.820

Is this correct? I double checked by finding that the applying the transformation matrix to w1 and w2 returned w1 and w2 which is what it should do. Thanks!

Alternatively, you can use the fact that $w_1/5$ and $w_2/\sqrt{50}$ form an orthonormal basis for W to see that T is given by the formula $Tx = \tfrac1{25}\langle x,w_1\rangle w_1 + \tfrac1{50}\langle x,w_2\rangle w_2$. The resulting matrix agrees with the one above.