Linear Algebra Math

Question

Let v1 = (1, -2, 3) and v2 = (-2, -2, 4).

Which of the following vectors are in span{v1, v2}?

(i) (3, 0, -1)

(ii) (5, 2, -4)

(iii) (8, 2, -5)

(A) (ii) and (iii) only (B) (iii) only (C) (i) and (ii) only (D) (ii) only (E) all of them (F) none of them (G) (i) only (H) (i) and (iii) only

Consider the vectors

u  =  [ 9

−1

−11  ,    v  =  −9 −1 −11  ,    w  =  9 1 −11 in ℝ3 True or False: The vector x1 x2 x3 is in span{u, v, w} for all x1, x2, x3 ∈ ℝ.

−11

Answer 1

(3, 0, -1) = v₁ - v₂ So, (3, 0 -1) is in the span of {v₁ , v₂}

If (5, 2, -4) is in the span, then (5, 2, -4) = a(1, -2, 3) + b(-2, -2, 4), for some numbers a and b.

So, a - 2b = 5, -2a - 2b = 2, and 3a + 4b = -4

From the second equation, a + b = -1, so b = -1 - a. Therefore, a - 2(-1 - a) = 5. Thus, a = 1 and b = -2. But then, 3a + 4b = -4 is false. So, (5, 2, -4) can't be in the span.

Similarly, if (8, 2, -5) is in the span, then (8, 2, -5) = r(1, -2, 3) + s(-2, -2, 4). This means that r - 2s = 8,

-2r - 2s = 2, and 3r + 4s = -5. So, from the second equation, s = -1 - r. Therefore, r - 2(-1 - r) = 8. So, r = 2 and s = -3. But then 3r + 4s = -5 does not check. So, (8, 2, -5) is not in the spanning set either.

Therefore, the answer is choice (G).