S13

Question 1

The least-squares solution of Ax = b is the point in the column space of A that is closest to b.

Answer 1

True

It is the point such that ∥Ax − b∥ is minimal, and this is exactly the distance from the point b to the column space of A.

Question 2

The least-squares solution of Ax = b is given by x^ = (A^T A)⁻¹A^Tb.

Answer 2

False

This is only true if the columns of A are linearly independent.

(Although the question is a bit ambiguous, because you could take the fact that we have written (A^TA)⁻¹ to mean that A^TA is invertible, in which case the columns of A are linearly independent, and so the statement is true.

We will avoid this kind of confusion on the exam.)

Question 3

If the columns of A are linearly independent, then Ax = b has a unique least- squares solution.

Answer 3

True

The formula from from the previous question in this series gives the unique solution in this case.

Question 4

If A = QR is a QR factorization of A, then the least-squares solution of Ax = b is x^ = R⁻¹Q^Tb.

Answer 4

False

Again, only true if the columns of A are linearly independent.

(As before, this is a bit ambiguous. In fact, we only defined QR factorization when A has independent columns.)

Question 5

If a matrix is orthogonal, then it is symmetric.

Answer 5

False

Question 6

If an n × n matrix is symmetric, then it has n distinct eigenvalues.

Answer 6

False

Question 7

A matrix is orthogonally diagonalizable if and only if it is nonsingular and symmetric.

Answer 7

False

A is orthogonally diagonalizable, but singular.