For each of the following.choose the correct option. If the answer is True, expl

Question

For each of the following.choose the correct option. If the answer is True, explain how you know If the answer is False, give a counter-example.(9 points) a. True or False: every matrix that has orthogonal columns is an orthogonal matrix b. True or False: every linearly independent set of vectors in gt' is an orthogonal set c. True or False: if two vectors vi and v2 are orthogonal, then they are also linearly independent d. True or False: if two vectors vi and v2 are orthogonal, and neither vi nor v2 equals zero, then vi and v2 are linearly independent e. True or False: if A is an orthogonal matrix, then A is invertible f. True or False: if A is an orthogonal matrix, then AAI g. True or False: if A is a matrix that has orthogonal columns, then A'A I h. True or False: if U has orthogonal columns, then U'U-D where D is diagonal. i. True or False: if A and B are each orthogonal matrices, then their product AB will also be an orthogonal matrix

Explanation / Answer

3 a. False. An orthogonal matrix is a square matrix with orthonormal columns.

b. False. A set in R3 is only orthogonal if every dot product between every two vectors is 0.

c. True. Orthogonal vectors are necessarily linearlly independent.

d. True. Orthogonal vectors are necessarily linearlly independent.

  e. True. If Q is an orthogonal matrix, then QT = Q-1.

  f. True. Then AT = A-1 so that AAT = AA-1 = I.

g. False.

h. False. If U is a 3x3 matrix with (1,1,1)T,(-1,0,1)T and (1,-2,1)T as columns, then UUT is not a diagonal matrix.

i. True. If AAT = AT A = I and BBT = BT B = I,then (AB)T(AB) = I

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