Question

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).]

Answer #1

9. (a). The vectors e_{1} = (1,0,0)^{T} and
e_{2} = (0,1,0)^{T} are 2 orthogonal vectors in
R^{3} that are linearly dependent.

(b). The vectors e_{1} = (1,0,0)^{T} ,
e_{2} = (0,1,0)^{T} and e_{3} =
(0,0,1)^{T} are 3 orthogonal vectors in R^{3} that
are linearly dependent.

( c). Let Q =

1 |
0 |

0 |
1 |

0 |
0 |

Then QQ^{T} =

1 |
0 |
0 |

0 |
1 |
0 |

0 |
0 |
0 |

Thus, the column vectors of Q are orthonormal and QQ^{T}
≠ I_{3}.

(d). It is not possible to have a 3x3 matrix Q with orthonormal
columns such that QQ^{T} ≠ I_{3} as such a matrix
is an orthogonal matrix for which, we necessarily have
QQ^{T} = I_{3}.

(e). It is not possible to have a 3x3 matrix Q_{1} and a
3x2 matrix Q_{2} with orthonormal columns such that
Q_{1}Q_{2} does not have orthonormmal columns. In
such a case, the columns of Q_{1}Q_{2} will be
necessarily orthonormal.

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P
Q
5
0
0
0
-1
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1
1
13
2
0
0
2
0
1
3
-6
4
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2
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1
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10
0
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i
1
2
3
4
5
6
Xi
2
6
3
1
1
9
Yi
13
17
12
12
9
22
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