Let F be any field, and let V be the vector space of all polynomials of degree at most d in the variable t with coefficients in F. Exhibit an explicit isomorphism between V and the column space F^{n} for suitably chosen n.
Let F be a field, and let F^{m}_{n} be the vector space of all m \times n matrices with entries in F. If V and W are vector spaces over F, then Hom(V, W) denotes the set of all F-linear maps from V to W. Hom(V, W) is itself a vector space over F under pointwise addition of linear maps and pointwise multiplication of a linear map by a scalar from F.
Define
| Phi : F^{m}_{n} -----> Hom(F^{n}, F^{m}) |
by defining Phi(M) to be the linear map for which
| (Phi(M))(X) = MX |
for each X \in F^{n}. Show that Phi is linear. (Proof of the linearity of Phi(M) for given M is not asked here.)
Let f: V -----> W be an injective linear map of vector spaces over the field F. Prove that if elements v_{1}, v_{2},..., v_{r} in V are linearly independent, then f(v_{1}), f(v_{2}), ..., f(v_{r}) are linearly independent elements of W.
Let F [t] be the vector space of polynomials in one variable t over the field F, and let D: F[t] --> F [t] be the map defined by *1*
| D |
| = SUM_{j}^{j-1}[ j c t ] . |
Show that D(f . g) = f . D(g) + D(f) . g.
Compute the kernel and image of D when F is the real field R.
Compute the kernel and image of D when F is the field F_{2} of integers mod 2.
How many rational scalars c are there for which the matrices
| and |
|