Advanced Linear Algebra Math 424/524

Assignment No. 2

Due October 16, 2002

1. Prove that if two matrices of the same size over a field are both in reduced row echelon form and are also row equivalent, i.e., each may be obtained from the other by a finite sequence of elementary row operations, then they must be equal.

2. Let R denote the field of real numbers.

(a) What is the linear subspace of R^{n} spanned by the set of columns x ∈ R^{n} having the property that every coordinate of x is non-zero?

(b) What is the linear subspace of the vector space M_{n}(R) of n × n real matrices that is spanned by the set of invertible n × n matrices?

3. Let F be a field, and let V denote the vector space F [t] of polynomials in the variable t with coefficients in F. Let a be a given element of F, and let U_{a} be the linear subspace of V consisting of all polynomials divisible by the polynomial t − a. Does the isomophism class of the quotient space V/U_{a} depend on the choice of a?

Hint: Consider the linear map s_{a} : F [t] → F defined by s_{a}(f) = f(a).

4. If F is any field, let V be the vector space M_{n}(F) of n × n matrices over F. For given A, B ∈ V define an F -linear endomorphism φ_{A,B} of V by

φ_{A,B}(M) = AMB .

(a) For what pairs A, B is the endomorphism φ_{A,B} equal to 0?

(b) Is every endomorphism of V equal to φ_{A,B} for some pair A, B?

5. Let X be a vector space over a field F, and let ψ be a linear map from X to X for which

ψ ∘ ψ = ψ .

Let V be the subspace of X that is the image of ψ, let j be the inclusion of V in X, and let q : X → V be the “projection” of X on V that yields the canonical factorization ψ = j ∘ q of ψ through its image.

Define a subspace U of X and a linear map p : X → U so that, with i : U → X the inclusion of U in X, one has the relations among i, j, p, and q characterizing an isomorphism of X with the Cartesian product U × V, i.e.,

pi = 1
pj = 0
qi = 0
qj = 1
ip + jq = 1