Question

If vectors v1, ..., vp span a subspace W and if x is orthogonal to each vj for j = 1, ..., p then xis in W's complement.

a. True
b. False

Answer 1

The statement is true, as a vector orthogonal to all vectors spanning a subspace W is not part of W and lies in W's orthogonal complement.

Step-by-step explanation:

The question asks whether it's true or false that if vectors v1, ..., vp span a subspace W and if vector x is orthogonal to each vj for j = 1, ..., p, then x is in W's complement. The statement is true. If x is orthogonal to each of the vectors that span W, then x cannot be expressed as a linear combination of these vectors, hence it lies outside of W. Therefore, x must lie in the subspace that is orthogonal to W, which is the complement of W.