All vectors are in R^n. Check the true statements below. A. Not every linearly i
ID: 3123128 • Letter: A
Question
All vectors are in R^n. Check the true statements below. A. Not every linearly independent set in R^n is an orthogonal set. B. A matrix with orthonormal columns is an orthogonal matrix. C. If L is a line through 0 and if cap y is the orthogonal projection of y onto L, then ||cap y|| gives the distance from y to L. D. 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. E. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.Explanation / Answer
A. Not every linear independent set in R n is an orthogonal set. TRUE .
It is only orthogonal if every dot product between two elements is 0.
B. A matrix with orthonormal columns is and orthogonal matrix. FALSE. It must be a square matrix
C 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||
D 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)
E 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