Show that the function (V1, V₂) = 2w1W2 + x1x2 - Y1Y2 +2122 defined for any vectors V₁ = (x1, y1, 21, W₁) and v₂ = (x2, Y2, 22, W2) in vector space R4 is not an inner product. Hint. Indicate which axiom(s) of inner product space is (are) not satisfied and give an example of the vector(s) which do(es) not satisfy the axiom(s).

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 = 35, v₂ · V2 = 334, V3 V3 . . = 16, w.v₁ = -105, w. v2 - 2338, w - V3 then w = =v₁+0₂ + v3. : 48, =

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.