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Flashcards in S12 Deck (5)
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1
Q

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

A

True

The a cancels out of the formula:

2
Q

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

A

True

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

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

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

3
Q

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

A

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 ATA = I) has det(A) = ±1, because 1 = det(I) = det(ATA) = det(A)2 implies det(A) = ±1.

4
Q

If ATA = I, then A must be square.

A

False

5
Q

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

A

True

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

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

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