Prove or give a counterexample: If v1, V2, .. Vm and w1, W2, .. Wm are linearly independent lists of vectors in a vector space V (over either Q, R, C), then 5v, + W1, V2 + W2, .. Vm + Wm is linearly independent.

a) Identify three non-zero vectors in span{v1,v2}. b) Show that {v1,v2} is linearly independent. c) Identify a vector u (u not equal to v1 or v2) such that {v1,v2,u} is linearly dependent.

The vectors 2 -0--0 V -2 -3+k 1 u: 3 -8 are linearly independent if and only if k # W 2

Suppose v1, v2, v3, 04, Ū5 and ve are vectors in R6 satisfying the following linear relation: 201 + 302 – Ủ4 + 205 - Which of the following must be true? (Select all that apply) span(v1, v2, v3, 4, 06) = span(71, v3, 74, 05, V6) span(v1, v2, v3, 74, 05) = span(v1, v2, v4, V5, 76) span(v1, v2, v3, v4, v6) = span(v1, 02, v3, v4, Ü5 , v6) span(v1, v2, V3, v6)= span(73, v4, V5 , V6) O span(v1, v2, v3, V4, 05) = span(72, 03, v4, 05, V6 )

6. Determine if the vectors vi = (1,–1,3, –1), v2 = (1, –1,4, 2), v3 = (1, –1,5, 7) are linearly dependent or 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 V1V1= 27, V2 V2 = 54, V3 V3 = 16, . . w.v₁ = -27, w v2 = 378, w V3: ₁+₂+ V3. then w = = 32, =