MTH207: 1. Consider the line in R3 Passing Through the Points (−5, 1,−2) and (−1, 0, 1): Linear Algebra Assignment, SUSS, Singapore

1. Consider the line in R³ passing through the points (−5, 1,−2) and (−1, 0, 1).

(a) Express the line in the form {x + w | w ∈ span (S)}, where x is a vector in R³ and S ⊆ R³.

(b) Express the line using the set notation in implicit form. You must justify your answer.

2. Consider the line in R³ passing through the points (−4,3,2) and (2,-1,1).

(a) (2 points) Express the line in the form {x + w | w ∈ span (S)}, where x is a vector in R³ and S ⊆ R³.

(b)Express the line using the set notation in implicit form. You must justify your answer.

3. (a) Suppose that U and V are both subspaces of Rⁿ. Prove that

U + V = {u + v : u ∈ U and v ∈ V} is also a subspace of Rⁿ .

(b) Write down the subspace U + V explicitly if

U = {(t, −3t, 5t) : t ∈ R} and V = {(0, 7t, 2t) : t ∈ R} .

(c) (3 points) Write down the subspace U + V explicitly if

U = {(t, 2t, 3t) : t ∈ R} and V = {(t, -2t, 0) : t ∈ R} .

4. Suppose that {v₁, . . . , v_k , v_k+1} is a set of linearly independent vectors in Rⁿ . Let u = v₁ + · · · + v_k + v_k+1. Prove that {v₁, . . . , v_k , u} is linearly independent.