All vectors are in R^n. Check the true statements below: If y is a linear combin
ID: 3032964 • Letter: A
Question
All vectors are in R^n. Check the true statements below: 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. A matrix with orthonormal columns is an orthogonal matrix. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. Not every linearly independent set in IR" is an orthogonal set. 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.Explanation / Answer
The statement is True. If y is a linear combination of the orthogonal set { u1 ,u2,…,un}, and if y = c1 u1 +c2 u2 + …+cn un , then ci = (y·ui)/( ui ·ui). The statement is False. An orthogonal matrix is also a square matrix which has not been stated here. The statement is False. Normalisation affects only the magnitude of the vector, and not their directions. The statement is True. For example (the set { (1,0)T , (1,1)T} in R2 is linearly independent, butt not orthogonal.. The statement is FALSE The distance is ||y – y’|| ( Here y’ is the orthogonal projection of y onto L; the other symbol is difficult to make).