Consider the vector space V = P₂ (R). That is, the space of all polynomials with degree less than or equal to 2. Consider the following two bases for V, B = {1+x,1-x, 3+x²} and C = {1, x, x²} Please do the following: • Consider the polynomial f(x) = -5 + 2x - 7² and find the component vector [f(x)]B • Find the change of basis matrix PB+C (don't confuse this with Pc+B !) • Verify by computation that PB-c[f(x)] = [f(x)]B Show complete reasoning and a clear conclusion to all questions. No determinants are allowed in your solution.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.