Linear Algebra Answers

find basis and dimension of the subspace W of V spanned by A = [[1 2] [-1 3]], B = [[2 5] [1 -1]], C = [[5 12] [1 1]], D= [[3 4] [-2 5]]

Let T be an element of L(R³) such that-4,5,"7" are it eigenvalues. show that T(x)-9x=(-4,5,"7" )

Let W = {(X1, X2, X3) €R³: X2 + X3 = 0}. How do I show that W is a subspace of R³? What are two subspaces W1 and W2 of R³ such that R³=W⊕W1and R³=W⊕W2 but W1≠W2?

Let A and B be n × n matrices. Prove that trAB = trBA and trA = trA^t

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Let V and W be n−dimensional vector spaces, and let T : V → W be a linear transformation.

Suppose β is a basis for V . Prove that T is an isomorphism if and only if T(β) is a basis for W.

et T : R3 —> R2 be given by : T (x1, x2, ,X3 = (x1 +x2 + X3, X2 + X3 ). Prove that T is a linear transformation. Also find the rank and nullity of T.