# Find the standard matrix of an orthogonal projection

• Apr 22nd 2010, 11:27 PM
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]

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:

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]

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