finding a matrix representation of a linear transformation

Answer explanation by Al Great observations! Let's address your main point first: Basis B for P3 in Part (b) You're correct: the given basis B =1-x,, 1 + 3x + 2x3,, 5 + 4x + 4x2 is not a full basis for P3, because P3, the space of all polynomials of degree at most 3, should have dimension 4 (basis should have 4 elements). Here, B only has 3 vectors, all without a terms. This is an issue if we are to map from all of P3 into P4, because to describe any element of P3, we need a 4- vector basis (e.g., 1, x, x2, 23). What does this mean for (b)? . As stated, basis B cannot span all of P3 because it's only 3 vectors (so the problem is either intentionally asking for a transformation from a 3-dimensional subspace of P3, or it is a typo/oversight). . If you proceed as stated, you must note to your instructor that you are working on the 3- dimensional subspace spanned by B, not all of P3. Part (a): Standard Bases for P3 and PA . Standard basis for P3: 1, x, a2, x3 . Standard basis for PA: 1, ac, ac2, ac 3, For L(p(x)) = x . p(x): Let p(x) = do + a12 + a2x2 + a323. L(p(x)) = x(do + a12 + a2x2 + a323) = dox + alx2 + a2x3 + aga Representing p(x) as [ao al a2 a3] (in P3), the matrix for L (from P3 to P4) is: 0 0 0 01 0 0 00 1 0 0 0 0 1 0 0 0 0 1 . This is a 5 x 4 matrix because it sends a 4-vector to a 5-vector. Part (b): Matrix for L with Nonstandard Bases . B = b1 = 1 - x2,, b2 = 1 + 3x + 2x2,, b3 = 5 + 4x + 4x2 (3 elements) . C =1, x, x2, 23, x (standard basis for PA) Steps: 1. Apply L to each basis vector in B: . L(b1) = x(1 -x2) = x -23 . L(b2) = x(1 + 3x + 2a?) = x + 3x2 + 2x3 . L(b3) = (5 + 4x + 4x2) = 5x + 4x2 + 423 2. Express each result as a column vector in basis C: . x - x3 [0, 1, 0, -1, 0]7 . x + 3x2 + 2x3 - [0, 1, 3, 2, 0] . 5x + 4x2 + 4x3 - [0, 5, 4, 4, 0]T 3. Form the matrix whose columns are these coordinate vectors: [0 0 01 1 50 3 4 - 1 2 40 0 0 This is a 5 x 3 matrix. This is a 5 x 3 matrix. Summary/Comment on Basis B . Properly, you should note: B does not span all of P3; any further computations involving the entire P3 will be incomplete or ill-defined. If this was not intentional in your course, you should mention this to your instructor. If you have questions about how this affects transformation properties, or if you'd like to explore what happens if a proper 4-vector basis is used instead, feel free to ask