, L(p(t)) = (t+1)p'(t) - 2p(t). As usual,
is the space of 2nd order polynomials in the variable t.
Problem 1: Consider the linear operator
,
L(p(t)) = (t+1)p'(t) - 2p(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-5x),
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.