Let V be a n-dimensional vector space over a field F, v be a basis of V, and w the dual basis of V^{*}. Let v' be another basis of V and w' the basis of V^{*} dual to it. If the change of basis between v and v', regarded as rows of vectors, is given by v' = v A for an n \times n matrix A and if also for V^{*} one has w' = w B, how is the matrix B related to the matrix A ?
Let f_{j} : V_{j} -----> W_{j} for j = 1, 2 be linear maps.
Explain briefly why there is a unique linear map
| f_{1} \otimes f_{2} : V_{1} \otimes V_{2} -----> W_{1} \otimes W_{2} |
for which
| (f_{1} \otimes f_{2})(v_{1} \otimes v_{2}) = f_{1}(v_{1}) \otimes f_{2}(v_{2}) whenever v_{1} \in V_{1}, v_{2} \in V_{2} . |
If for chosen bases in V_{j}, W_{j}, the linear map f_{j} has the matrix
| for j = 1, 2 , |
Let R denote the field of real numbers. What more familiar description may be used to describe the d^{th} symmetric power S^{d}((R^{2})^{*}) of the dual space of the real plane R^{2} to a student who has completed the calculus sequence?
Let Q denote the field of rational numbers, P(t) the polynomial t^{5} + 5 t^{4} + 6 t^{3} + 7 t^{2} + 8 t + 9 in Q [t], and let
| V = Q [t] / P(t) Q [t] . |
Prove that V has dimension 5 over Q by exhibiting a basis of V.
If pi denotes the quotient map from Q [t] to V, and tau denotes the linear endomorphism of Q [t] defined by tau(F(t)) = t F(t) for F \in Q [t], explain briefly why there is a linear endomorphism phi of V such that phi \circ pi = pi \circ tau.
Find the matrix of phi with respect to the basis of V determined in response to the first part of this exercise.
What is the characteristic polynomial of the matrix found in the previous part of this exercise?
Let V and W be vector spaces over a field F. There is a natural linear map
| rho : V \otimes W -----> Hom(V^{*}, W) |
| rho(v \otimes w) = |
| . |