All vectors are in R^n. Check the true statements below: If the vectors in an or
ID: 3033047 • Letter: A
Question
All vectors are in R^n. Check the true statements below: 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. A matrix with orthonormal columns is an orthogonal matrix. 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. Not every linearly independent set in R^n is an orthogonal set.Explanation / Answer
A. The statement is False. Normalizing a vector changes only its magnitude, but not its direction.
B. The statement is False. If L is a line through 0, and if y’ is the orthogonal projection of y onto L, then the distance from y to L is ||y – y’||.
C. The statement is False. An orthogonal matrix is also a square matrix apart from having orthonormal columns.
D. The statement is True. If x = c1 u1 +c2 u2 +…+cn un +…, then ci = (y·ui)/( ui· ui).
E. The statement is False. The set of vectors v1 = (1,1,0)T and v2 = (0,1,1)T in R3 in a linearly independent set but this set is not orthogonal.