Advanced Linear Algebra Math 424/524

Assignment No. 1

Due October 2, 2002

1. 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.

2. Let F be a field, and let F_{n}^{m} be the vector space of all m × 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

Φ : F_{n}^{m} ⟶ Hom(F^{n}, F^{m})

by defining Φ(M) to be the linear map for which

(Φ(M))(X) = MX

for each X ∈ F^{n}. Show that Φ is linear. (Proof of the linearity of Φ(M) for given M is not asked here.)

3. 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.

4. 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 (∑ c_{j}t^{j}) = ∑ jc_{j}t^{j−1} .

(a) Show that D(f · g) = f · D(g) + D(f) · g.

(b) Compute the kernel and image of D when F is the real field R.

(c) Compute the kernel and image of D when F is the field F_{2} of integers mod 2.

5. How many rational scalars c are there for which the matrices

( \begin{matrix}c & 0 \\ 0 & c \end{matrix} ) and ( \begin{matrix}c & 1 \\ 0 & c \end{matrix} )

are similar? Justify your answer.

1In this expression j as an index is an integer. How does one interpret jc_{j} given that c_{j} is in F? As long as j is a non-negative integer, the meaning of jc_{j} is “c_{j} added to itself j times in the field”. If j < 0, then jc_{j} is understood as the negative of (−j)c_{j}. Consistent with that j itself can be interpreted in F as j · 1 where 1 denotes the multiplicative identity of F. Thus, in F_{2}: j = 0 if j is even, while j = 1 if j is odd.