Linear Algebra dealing with orthonormal basis

MATH 310 Homework due 07/26/2013

1. Show that the vectors

(1/(3 √ 2), 1/(3

√ 2),−4/(3

√ 2))T , (2/3, 2/3, 1/3)T , (1/

√ 2,−1/

√ 2, 0)T

form an orthonormal basis of R3. Find the coordinates of the vector (1, 4, 3)T with respect to that basis.

2. Find an orthonormal basis of the image of the linear map: T : R2 → R3 with

T((x, y)T ) = (3x−2y, x + y, x−y)T

3. Apply the Gramm-Schmidt orthogonalization process to the vectors (1, 3, 2)T and (1, 0, 1)T in order to get an orthonormal basis of the subspace that they span.

4. Find an orthonormal basis of the kernel of the linear map T : R3 → R with T((x, y, z)T ) = x−3y + z.

5. Find an orthonormal basis of the subspace:

V = {(x, y, z, w)T : x + y + z + w = 0}

of R4.

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