Question

All vectors and subspaces are in RnR n. Mark each statement True or False. Justify each answer.

a. Not every linearly independent set in RnR n is an orthogonal set.
b. 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.
c. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
d. A matrix with orthonormal columns is an orthogonal matrix.
e. If L is a line through 0 and if ˆyy^ is the orthogonal projection of y onto L, then ∥ˆy∥∥ y^ ∥ gives the distance from y to L.

Answer 1

a. True

b. True

c. False

d. False

e. False

Explanation:

The matrix with orthonormal column must be in square for that it forms orthogonal matrix. Every orthonormal set of non zero vector is linearly independent. This statement is correct because there is linear combination of non zero vectors.