S12

Question 1

The orthogonal projection of u onto v is the same as the orthogonal projecton of u onto a v for any a ≠ 0.

Answer 1

True

The a cancels out of the formula:

Question 2

If W is a subspace of ℝⁿ and u ∈ W, then proj_W(u) = u.

Answer 2

True

By Theorem 8 in the book, there is a unique decomposition

u = p + o with p = proj_W(u) ∈ W and o ∈ W⊥.

But if u ∈ W, then u = u+0 is such a decomposition, hence proj_W(u) = u.

Alternatively, there is an orthogonal basis of W that includes u. Applying the formula for the orthogonal projection with this basis will exactly give u.

Question 3

Let A be an n × n matrix. The columns of A form an orthonormal basis of ℝⁿ if and only if det(A) = 1.

Answer 3

False

The matrix A (image) has orthonormal columns but determinant -1, and

the matrix B = ([1 0], [1 1]) has determinant 1 but does not have orthogonal columns.

So the implication doesn’t work in either direction.

However, it is true that an orthogonal matrix (a matrix with A^TA = I) has det(A) = ±1, because 1 = det(I) = det(A^TA) = det(A)² implies det(A) = ±1.

Question 4

If A^TA = I, then A must be square.

Answer 4

False

Question 5

A square matrix has orthonormal columns if and only if it has orthonormal rows.

Answer 5

True

By Theorem 6 in the book, a matrix A has orthonormal columns if and only if A^TA = I.

But then A⁻¹ =A^T, so we also have AA^T = I, or in other words (A^T )^T A^T = I.

By the same theorem, this means that A^T has orthonormal columns, which means that A has orthonormal rows.