UCSD ECE 269 Handout

1. When is it true? (5 points for each correct answer, -3 point for each wrong answer, 0 point for each blank) Fill in each blank with “always,” “sometimes,” or “never.” For example, • A nonsingular matrix is always invertible. • A square matrix is sometimes full-rank. • A strictly tall matrix is never onto. Here the matrix dimensions are such that each expression makes sense, but they are otherwise unspecified. Every vector and matrix has real entries. (a) The union of two subspaces of R is sometimes a subspace. Solution: Consider two subspaces V and W of R such that W ⊆ V. In this case, V ∪ W = V which is a subspace. On the other hand, consider the two subspaces V ′ and W ′ of R spanned by the vectors v = [