1-i = [2+2 Exercise 2.2.3 Let c₁ = 2i, c₂ = 1 + 2i, and A = 3 2+2 4+1 Verify Properties (vi) and (viii) in showing C2x2 is a complex vector space.

28. Is C a real vector space? Explain.

Determine if the vector product <A,B> = a0b0 defines a valid inner product on P2(R) for any A=a0+a1x+a2x^2 and B=b0+b1X+b2x2. Explain your answer clearly.

Let S be the parallelogram then determine the area of parallelogram by the vectors [h | 1 and let A = h and [0 h+1] h=102 Also compute the area of the image of parallelogram under the linear transformation ""

7. Find the coordinates of the vector v relative to the ordered basis B. a) B = {x² + 2x + 2, 2x + 3,−x² + x + 1} v=-3x² + 6x + 8 14 D b) B = {[1₁¹] [ = (₁ 3²1 -CAC) = B₂ -QAED = 0 v= c) B₁ v= 22

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"On R², define the operations of addition and scalar multiplication as follows: (X₁, X₂) + (y₁ Y₂) = (x₂ + y₂ +1, x₂ + y₂ + 1), c(x₁, x₂) = (Cx₂ + C-1, ₂+c-1). Then R² with these operations forms a vector space." Note that here the zero vector is (-1,-1) and the additive inverse of (x1, x2) is (-X₁-2, -x₂-2), not (-x₂-x₂)- a) Show that (-1, -1) acts as the zero vector under these definitions.