Question

3. In R⁴ , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R⁴? In other words, can you write down any vector (a, b, c, d) ∈ R⁴ as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ?

4. In the vector space P₂ of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span of the set {x + 2, x² − 1}.

5. Let S be the subspace of P₂ defined by S := {ax² + bx + 2a + 3b : a, b ∈ R}, for different choices of real numbers a and b (you don’t need to show here that S is indeed a subspace, and can assume. But is a good practice problem). Find a basis, and hence dimension for S.

Answer 1