Question

9. Either give an example of each of the following or explain why it would be...

9. Either give an example of each of the following or explain why it would be impossible.

(a) [2 points] Two orthogonal vectors in R 3 that are linearly dependent.

(b) [2 points] Three orthonormal vectors in R 3 that are linearly dependent.

(c) [2 points] A 3 × 2 matrix Q whose column vectors are orthonormal and QQT 6= I.

(d) [2 points] A 3 × 3 matrix Q whose column vectors are orthonormal and QQT 6= I.

(e) [2 points] Two orthogonal matrices Q1 (3 × 3) and Q2 (3 × 2) such that Q1Q2 is not an orthogonal matrix. [Note: A matrix Q is said to be orthogonal if QTQ = I, i.e., the columns of Q are orthonormal (not orthogonal).]

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Answer #1

9. (a). The vectors e1 = (1,0,0)T and e2 = (0,1,0)T are 2 orthogonal vectors in R3 that are linearly dependent.

(b). The vectors e1 = (1,0,0)T , e2 = (0,1,0)T and e3 = (0,0,1)T are 3 orthogonal vectors in R3 that are linearly dependent.

( c). Let Q =

1

0

0

1

0

0

Then QQT =

1

0

0

0

1

0

0

0

0

Thus, the column vectors of Q are orthonormal and QQT ≠ I3.

(d). It is not possible to have a 3x3 matrix Q with orthonormal columns such that QQT ≠ I3 as such a matrix is an orthogonal matrix for which, we necessarily have QQT = I3.

(e). It is not possible to have a 3x3 matrix Q1 and a 3x2 matrix Q2 with orthonormal columns such that Q1Q2 does not have orthonormmal columns. In such a case, the columns of Q1Q2 will be necessarily orthonormal.

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