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.
Let R denote the field of real numbers.
What is the linear subspace of R^{n} spanned by the set of
columns x \in R^{n} having the property that every coordinate
of x is non-zero?
What is the linear subspace of the vector space M_{n}(R)
of n \times n real matrices that is spanned by the set of
invertible n \times n matrices?
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).
If F is any field, let V be the vector space M_{n}(F)
of n \times n matrices over F. For given A, B \in V define
an F-linear endomorphism phi_{A,B} of V by
For what pairs A, B is the endomorphism phi_{A,B} equal
to 0 ?
Is every endomorphism of V equal to phi_{A,B} for some
pair A, B ?
Let X be a vector space over a field F, and let psi
be a linear map from X to X for which
Let V be the subspace of X that is the image of psi, 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 psi = j \circ q of psi 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 \times V, i.e.,