 # 1. (10 points) Decide which of the following functions D from the 3...

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1. (10 points) Decide which of the following functions D from the 3 x3 matrices to R is a multilinear function: (a) D(A) = @11 + @2 + @33 (b) D(A) = (a11)² + 3a11@22 (c) D(A) = 0. (d) D(A) = 2a11@12@13, 2. Let V be a vector space over R and let A € R. Let T: V V be a linear transformation. Prove that the set Sx = {v € V | T(v = Av} is a vector subspace. (A non-zero vector V verifying T(v) = Av is called an eigenvector with respect to X which is an eigenvalue). 3. Assume that V is a vector space, T : V V a linear map. Assume that T(V1) = A1V1 and T(V2) = À2V2, where A1 # À2- Prove that V1 and V2 are linearly independent 4. Let T:V V be a linear map such that Tn = 0. Prove that the only possible eigenvalue is 0. 5. Let T : V V prove that V = ker T + ImT. (Recall that given two subspaces of V, say W1 and W2, W1 + W2 is the subspace {W1 + W2 I Wi € W}. 6. Let T : V V such that T is linear and one-to-one. Prove that if V1, , Vn are linearly independent so are T(V1), T(Vn).

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