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!

Quote:

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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!

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Alternatively, you can use the fact that and form an orthonormal basis for W to see that T is given by the formula . The resulting matrix agrees with the one above.