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1. (a) Define that T U - V is a linear map from the vector space U to the vector space V. (b) Assume that are dependent in U. Prove that T(r)) T(x) are linearly dependent in V. 2. (a) Let T:U - V be linear and W be a subspace of V. Prove that '(W) = X = {=|T(a) E w} is a subspasce of U. (b) Prove that dirn(X) dim(W). 3. State the definition of the null space and of the range of a linear map T. 4. Let U be a vector space of dimension 2n, that is, U has a basis v1, 02, Wn- Let T be the linear map T:U - U such that - - and 0,1=1, n. What can you say about dim(null-space(T)) and din(range(T))? Prove that null-space(T) = range(T). 5. Prove that the solution space of a linear homogemeous system AX = 0 of m equations in n unknowns with real coefficients is a subspace of R". 6. Let T: R - R™ be linear where m < n. Prove that the null-space of T is nom-trivial, that is different from {0}. 7. Prove that a linear map T:U - V is injective if and only if the null-space is trivial. 8. Let T be as linear map from R= to Rm. How is the matrix A for T with respect to the unit vectors defined? Write down the matrix for the linear map T in problem 4. 9. Express mul-space(T) and range(T) in terms of its matrix A. Relate the dimensions of these spaces to row-rank and column-rank of A. 10. Let T be a linent map from R= - R and let A € R (a) Define that A is an eigenvalue of T. (b) Define that E2 is the eigenspace for &

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