Lev V ={x 1 ,x 2 ,x 3 }l x 1 -2x 2 +x 3 =0 in R 3 1) find a basis for V 2) use G

Question

Lev V ={x1,x2,x3}l x1 -2x2 +x3=0 in R3

1) find a basis for V

2) use Gram-Schmidt process to find an orthonormal basis for V

3) combine the basis from 2) with a basis for Vperp to get an orthogonal basis for R3

4) use the basis from 3) together with the change of basis formula to find the matrix R for the linear transformation that rotates all vectors in R3 about the line Vperp by angle theta in the counterclockwise direction when view from above the plane V

5) confrim that the columns of R are orthonormal

Explanation / Answer

1) basis of v is (t1,(t1+t2)/2, t2) i.e (1,1/2,0), (0,1/2,1)

2) orthogonal (1,0,1) (1,1,-1)

3) {(t1+t2),t2, t1-t2}

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