(a) (6 marks) The set U={(x2−x)p∣p∈P3​} is a subspace of P5​ (You don't need to show this). Find a basis for U and state

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A 6 Marks The Set U X2 X P P P3 Is A Subspace Of P5 You Don T Need To Show This Find A Basis For U And State 1 (36.14 KiB) Viewed 37 times

(a) (6 marks) The set U={(x2−x)p∣p∈P3​} is a subspace of P5​ (You don't need to show this). Find a basis for U and state dim(U). Verify that your choice of basis is in fact a basis for U, that is, verify that it is a linearly independent spanning set for U. (b) Let W={B∈Mn​(R)∣AB=BA}, where A is a fixed, but unknown, matrix in Mn​(R). - (5 marks) Prove that W is a subspace of Mn​ (R) by showing that properties (i), (ii), and (iii) of the Subspace Criterion hold, that is, show that it is non-empty and is closed under both addition and scalar multiplication. - (6 marks) Now let A=[11​11​] so that W={B∈M2​(R)∣AB=BA}. Find a basis for W and state dim(W). Verify that your choice of basis is in fact a basis for W, that is, verify that it is a linearly independent spanning set for W.