For each of (a)-(d) below, decide whether the statement is Trueor False. If True, explain why. If False, give acounterexample.
(a) Two non-parallel planes in R 3 will always meet in aline.
(b) Let v, x1, x2 ∈ R 3 with x1 6= x2. Define two lines L1 andL2 by
L1 : (x, y, z) = x1 + sv, s ∈ R,
L2 : (x, y, z) = x2 + tv, t ∈ R.
Then the lines L1 and L2 will never intersect.
(c) Let v1, v2, v3 be three non-zero vectors in R 3 . Then 0(the zero vector) is a linear combination of v1, v2 and v3. (d) IfA and B are two 2 × 2 matrices, and AB = 02×2 (where 02×2 is the 2× 2 zero matrix), then at least one of A or B must be the zeromatrix.
