## Transcribed Text

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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