Consider two bases B = {b1,b2, b3} and C = {C1, C2, C3} for a vector space V such that b₁ = C₁ + 6c3, b2 = C₁ + 3c2₂ - C3, and b3 = 5c₁ C2. Suppose X = b1 + 4b2 + b3. That is, suppose [X]B O [10] 11 [10] 11 10 [7] 12 1 O [10 17 = 4 . Find [x]c Jhry

25. Consider the following elements of the polynomial vector space. Are they linearly independent? 1 + x,x+x², 1-x²

= {W₁, W2, W3} is a basis for R³, where 1 3 -3 --0--0--0 = W2 = W3 = -2 -4 1 Determine if the set B

Let S {V1, V2, V3} and T = {w₁, W2, W3} be two bases in the vector space P₂ {a+bx+cx²: a, b, c = R}, where V₁ = 1+x+3x², v₂ : −2 − 2x − 7x², V3 = 1 + 2x + 6x² and = : 1 + x − x², w₂ = W1 = 1+ 2x², W3 = 3-x+5x2 Find the transition matrix from T to S. Let v = 201 - 3v2 + V3, find the coefficients (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. =