Consider the set of vectors {V1, V2, V3} = = {A·A·A) following gives a geometric interpretation of span{V1, V2, V3}? O span{V1, V2, V3} is a straight line passing through the origin. O span{V1, V2, V3} is a plane passing through the origin. Ospan{V1, V2, V3} is the whole 3-dimensional space. O span{V1, V2, V3} is {0}. Which one of the

Let v₁ = (1,3,5), v² = (-2, 1, 7) and let P be the plane spanned by v¹ and v². (That is, P = (v1, v2).) Let √3 = (7,7,1). Show that (v1, v2, v³) as follows: First show that is in P. • Next explain why (v1, v2, v3) is a subspace of P. • Next explain why P = (v1, v2) is a subspace of (v1, v2, v3). • Conclude that (v1, v2, v3) is the exact same set as P.

Consider the set of vectors S= {u₁, U₂) in R³, where u₁ = (1,-1,2) and u₂ = (-2,0,3). Is the vector b= (12,-2, -11) a linear combination of the vectors in the set S? If it is, write b as a linear combination of u, and u₂. If it is not, explain why. Show all necessary calculations. Does the set S span R³? I in R², where Given the linear transformation T: R² R³ defined by T(x) = Ax for all x A = [u₁ U₂] 1-2 0 3 -1 2 does T map R² onto R³? Circle your choice below and briefly justify your answer. Circle One: [Onto / Not onto

Find three vectors v1, v2, v3 such that (1) any two span a plane, and (2) Span{v1, v2, v3} is also a plane. v1 = v2 = V3 =

Q5) Describe the Null A by listing the vectors that span the Null space. 135 0 A= 014 -2