**Problem 1:** Consider the linear operator
,

*L*(*p*(*t*)) = (*t*+1)*p*'(*t*) - 2*p*(*t*).
As usual,
is the space of 2nd order polynomials in the variable *t*.

a) What is the matrix of *L* relative to the basis
?

b) Find the dimension of the kernel of *L* and the dimension of
the range of *L*.

c) Find a basis for the kernel of *L*. Also, find a basis for
the range of *L*.

**Problem 2** This problem concerns changing bases in
. Let *S* be the standard basis. Let
be another basis, where
,
,
.

a) Find a matrix *P* that converts from the *T* basis to the
*S* basis. That is, so that for any vector *X*,
.

b) Find a matrix *Q* that converts from the *S* basis to the
*T* basis. That is, so that for any vector *X*,
.

c) There is a linear operator *L* whose matrix, relative to the
*S* basis, is
. Find the matrix of *L* relative to the *T* basis.

**Problem 3.** Let

a) What is the rank of *A*?

b) Find a basis for the null space of *A*.

c) Find a basis for the row space of *A*.

**Problem 4.** Let *V* be the span of
and
, where
,
and
.

a) Find an orthogonal basis for *V*.

b) Find an orthonormal basis for *V*.

**Problem 5.** True of False

a) The map
, *L*(*x*,*y*)=(*x*+*y*,*y*-1), is a linear
transformation.

b) The map , , is a linear transformation.

c) The map
, *L*(*x*,*y*)=(3 *x*+ *y*,*y*-5*x*),
is a linear transformation.

d) Let *L* be a linear transformation from
to
. If the rank of *L* is 3, then the kernel of *L* is 1-dimensional.

e) Let *L* be a linear transformation from
to
. If the kernel of *L* is 2 dimensional, then *L* is onto.

f) Let *L* be a linear transformation from
to
. If *L* is 1-1, then *L* is onto.

g) If a set of vectors is orthonormal, then it is a basis.

h) If a set of vectors is orthonormal, then it is linearly independent.

i) If a matrix is wider than it is tall (e.g. a matrix), then the row rank is greater than the column rank.

j) Let *L* be a linear transformation from
to itself. The solutions (in
) to the equation *L*(*p*)=0 form a vector space.