Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.
Let f be the Z-linear map from Z^{4} to Z^{3} defined by f(x) = M x for all x \in Z^{4} where M is the 3 \times 4 matrix
M = |
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Represent the cokernel of f as a direct sum of cyclic groups.
Find one matrix in each of the similarity classes of matrices over the field C (of complex numbers) that share the characteristic polynomial t^{3} - t^{2} - t + 1.
Let A be the 5 \times 5 matrix in the field F_{2} = Z/2Z
A = |
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Find a direct sum of companion matrices that is similar to A.
Let M be the 6 \times 6 rational matrix
M = |
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Find the sequence of successively divisible invariant factors as well as the minimal and characteristic polynomials of M.
Let V and W be n-dimensional vector spaces over a field F, and let b : V \times W --> F be F-bilinear. The bilinear form b determines a linear map phi_{b} \in Hom_{F}(V, \v{W}), with \v{W} = Hom_{F}(W, F), that is defined by
phi_{b}(v) = |
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Show that phi_{b} is an isomorphism if and only if for given bases v_{1}, …, v_{n} of V and w_{1}, … w_{n} of W one has det
( | b(v_{i}, w_{j}) | ) |
Produce such a bilinear form in the case where V = Lambda^{p} U and W = Lambda^{k-p} U when U is a k-dimensional vector space over F.
To what extent may one generalize the two preceding items to correct statements when F is a commutative ring, V, W free F-modules of rank n, and U a free F-module of rank k?