W4 Pick any vector space V that you want, as long as dim(V) ≥ 6. Then, pick 4 vectors in the space, say W = {w₁, w₁, W3, w}} that have the following property: 1. Applying the plus/minus theorem, you can pick two vectors from W, say W' = {wi, w2} so that span(W') = span(W). Prove, using the minus part of the plus/minus theorem that your vectors meet the above conditions. Then, explain in words how you created your set of vectors so that this worked out.

Q # ll Assume that Vy = (Q,-3,-3) and V2 = (3,1,-2).- You have been giveh the following vectors in Span { Vý , Vz}: There more than below. which of them be are may one correct auswer. o (-1, -4,-1) (-4,-5)1) ®(-5, -9,1)

12. Consider the vectors v1 = (2, -1, 5), v2 = (1, 3, -4), V3 = (-3, –9, 12) in R³. (a) Show that {v1, v2, V3} is linearly dependent. (b) Is vị E span{v2, v3}? Draw a picture illustrating your answer.

Let U be a subspace of R", and let S = { ū, v, w } be a set of three non-zero vectors in R”. Fill in the blanks so that the following statements are true. If you believe you don't have enough information to conclude anything for any of the cases, choose the answer "???". Note that there is no intended relationship between the parts. In each case, assume only what the statement says to assume. If SCU, then dim(U) > (Enter the largest integer that makes the statement necessarily true.) If S is linearly independent and S C U, then dim(U) 3. If S' is a spanning set for U, then dim(U) 3. If span(S) = U then S If S is a basis for U, then S linearly independent. linearly independent.

Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V₁, V2, V3) such that V₁ V₁ = 42, V₂ · V₂ = 542, V3 V3 = 36, -126, w - V₂ - 2710, w - V3 w. • V₁ then w= v₁+ = = = -36, V2+ V3.

Let v1 = <0, 1>, v2 = <-1, 0>, w = <-7, 1> vectors ∈ R2. Does the set {v1, v2} span R2?

Use the function to find the image of v and the preimage of w. T(V1, V2, V3) = (4V2 - V1, 4V, + 5v,), v = (1, –4, -3), w = (5, 22) (a) the image of v (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.)