TRUE OR FALSE and explain why In R5 there is a non zero subs

TRUE OR FALSE and explain why.

In R^5 there is a non zero subspace V whose dimension equals the dimension of its orthogonal complement V T or F?

Solution

FALSE:

Reason: Let S be a subset of an inner product space V . Then every vector of S is orthogonal to every vector of Span (S).

Let A be an m × n real matrix. Then Nul (A) and Row (A) are orthogonal complements of each other in Rⁿ, i.e., Nul (A) = [Row (A)] , and [Null (A)]= Row (A). Thus, Nul (A) + Row (A) = [Null (A)] + [Row (A)] = Dim Rⁿ = n.

Therefore if the subspace V, and its orthogonal complement V have equal dimension then n must be even.

But here n = 5 is not even.