All vectors are in Rn Check the true statements below If the

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.

Solution

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 +…+c_n u_n +…, then c_i = (y·u_i)/( u_i· u_i).

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.