S11

Question 1

Let u, v, w ∈ ℝⁿ. If u is orthogonal to v and v is orthogonal to w, then u is not orthogonal to w. True or False?

Answer 1

False

Take for instance the standard basis e₁, e₂, e₃.

(It is true in ℝ², though.)

Question 2

If the distance from u to v equals the distance from u to −v, then u and v are orthogonal.

T/F ?

Answer 2

True, if :

∥u − v∥ = ∥u − (−v)∥ = ∥u + v∥

then
(u − v) · (u − v) = (u + v) · (u + v)

⇒ u·u − 2u·v + v·v = u·u + 2u·v + v·v

⇒ − 2u · v = 2u · v
⇒ u·v = 0

Question 3

For a square matrix A, vectors in Col(A) are orthogonal to vectors in Nul(A).

Answer 3

False

Take for instance A and v = (1, −1) .

Then the inner product of v with a column (1, 2) equals −1, so they are not orthogonal.

It is true for the row space Row(A), however. A vector v is in Nul(A) if Av = 0,

which is the same as the inner product of v with each row of A being 0.

Question 4

Let W be a subspace of ℝⁿ. If x ∈ ℝⁿ is orthogonal to every vector in a basis of W, then x ∈ W^⊥.

Answer 4

True

If B is a basis of W , then B⊥ =Span(B)⊥ =W⊥