The following statements refer to vectors in Rn (or Rm) with the standard inner product. Mark each statement True or False. Justify each answer.
a. The length of every vector is a positive number.
b. A vector v and its negative -v have equal lengths.
c. The distance between u and v is ||u - v||.
d. If r is any scalar, then ||rv|| = r||v||.
e. If two vectors are orthogonal, they are linearly independent.
f. If x is orthogonal to both u and v, then x must be orthogonal to u - v.
i. The orthogonal projection of y onto u is a scalar multiple of y.
j. If a vector y coincides with its orthogonal projection onto a subspace W, then y is in W
k. The set of all vectors in Rn orthogonal to one fixed vector is a subspace of Rn
l. If W is a subspace of Rn, then W and W? have no vectors in common.
m. If {v1, v2, v3} is an orthogonal set and if c1, c2, and c3 are scalars, then {c1v1, c2v2, c3v3} is an orthogonal set.
n. If a matrix U has orthonormal columns, then U UT = I .
o. A square matrix with orthogonal columns is an orthogonal matrix.
p. If a square matrix has orthonormal columns, then it also has orthonormal rows.
q. If W is a subspace, then ||projW v||2 + ||v projW v||2 = ||v||2 .
r. A least-squares solution of Ax = b is the vector A in Col A closest to b, so that ||b - A || ≤ ||b - Ax|| for all x.
s. The normal equations for a least-squares solution of Ax = b are given by = (ATA)-1 A T b.