All vectors are in R^n. Check the true statements below: A matrix with orthonorm

Question

All vectors are in R^n. Check the true statements below: A matrix with orthonormal columns is an orthogonal matrix. Not every linearly independent set in R^n is an orthogonal set. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. If L is a line through 0 and if y is the orthogonal projection of y onto L, then ||y|| gives the distance from y to L. 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.

Explanation / Answer

A matrix with orthonormal columns is and orthogonal matrix. FALSE. It must be a square matrix

so option A is false

Not every orthogonal set in R n is linearly independent. FALSE Orthogonal implies linear independence.

option B is false

If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. FALSE Normalizing just changes the magnitude of the vectors, it doesn’t affect orthogonality.

option C false

If L is a line through 0 and if by is the orthogonal projection of y onto L, then ||by gives the distance from y to L. FALSE The distance is ||y by||

option D is false

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. TRUE cj = y·uj /uj.uj .

option E si true

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