Determine whether the following are vector subspaces of the

Determine whether the following are vector subspaces of the domain of the linear map x rightarrow Ax. If yes, explain. If no, provide a concrete counterexample. The solution set of Ax = b The kernel of A Determine whether the matrices [2 1 0 3] and [2 0 0 3] are similar. Explain. Let L be the plane x_1 + x_2 + x_3 = 0 and let v = [1 1 1]. Define a linear transformation T of L by Tx = x times v. What is the determinant of T? Find all eigenvalues and the corresponding eigenvectors of A = [1 4 2 -1] Reflection of the plane with respect to the main diagonal. Suppose A = [3 2 6 -1]. It can be shown that A has eigenvalues -3 and 5 with corresponding eigenvectors Solve the dynamical system x(n + 1) = Ax(n) with initial condition x(0) = [3 -1]

Solution

Solution of question (1) :

A linear map always confirms the operation of addition and scalar multiplication. And in case of Ax=B, both operations hold as determinant of A for inverse A, is a scaler that is further multiplied with Adj A to get inverse A that is required to get x. Thus it is scalar multiplication there as well as corresponding elements are added also while multiplying inverse A and matrix B, so here it is addition also.

So YES. the solution set of Ax=B is a vector subspace.

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Now as the kernel of a linear transformation is the set of all vectors v , so that

L(v)=0 or in other words kernel is only the solution set of homogenouous system of equation but in given case, it is not homigenous. Thus kernel of A is not a vector subspace.

This is answer of part (b)