A Matrix With Orthonormal Columns Is An Orthogonal Matrix.

Question: Not every linearly independent set in Rn is an orthogonal set.
Answer: True - A set is only orthogonal if every dot product between its two elements is 0.
{ [ 8 ] , [ 1 ] }
[ 3 ] [ 10 ]
is linearly independent but not orthogonal.
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Question: If y is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.
Answer: True - The weights can be computed by the equation cj = y . u j

uj . uj
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Question: If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
Answer: False - Normalizing just changes the magnitude of the vectors. Normalization does not affect orthogonality.
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Question: A matrix with orthonormal columns is an orthogonal matrix.
Answer: False - The matrix must be SQUARE, but the statement does not include this.
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Question: If L is a line through 0 and if yhat is the orthogonal projection of y onto L, then ||yhat|| gives the distance from y to L.
Answer: False - The distance from y to L is ||y - yhat||.
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Question: Not every orthogonal set in Rn is linearly independent.
Answer: False - Orthogonality indicates linear independence.
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Question: If a set S = {u1,..., up} has the property that ui . up = 0 whenever i != j, then S is an orthonormal set.
Answer: False - There may be a vector that does not have a magnitude of 1.
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Question: If the columns of an m x n matrix A are orthonormal, then the linear mapping x -> Ax preserves lengths.
Answer: True - by Theorem 7 (a) - ||Ax|| = ||x||
where A is an m x n matrix with orthonormal columns.
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Question: The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c != 0.
Answer: True - p 342 first paragraph
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Question: An orthogonal matrix is invertible.
Answer: True - Orthogonality indicates linear independence, which indicates invertibility, by the Invertible Matrix Theorem.
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