Linear Algebra dealing with orthonormal basis

Need 4 out of the five questions done with complete work shown short notice but the HW is do able in the alloted time I would need it in the next 5 hours or so.

 

Problem number two is already solved so don’t do that problem.

MATH 3

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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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