Coordinates and Linear Transformations
October 22, 2015

p~(1)
. In words, T
p~(2)
maps a polynomial in the variable t to the column vector of its values at t = 1 and t = 2.
For example,



1
1
0
3
T (1) =
, T (t) =
, T ( 2 + 3t t ) =
.
1
2
4
Consider the linear transformation P3
(a) Compute T (t2 ), T (t3 ), and T (3
T
/
R
2
defined by T (~p) =
t + 2t2 + 5t3 ).
In order to understand T as completely as possible, we will make use of coordinates.
[ ]B
/ R4 the coordinate mapping,
Let B = {1, t, t2 , t3 } be the standard basis for P3 and P3
which just maps a polynomial to the column vector of its coefficients. For example,
2 3
2 3
2 3
1
0
2
6
7
607
617
7 = ~e1 , [t]B = 6 7 = ~e2 , [ 2 + 3t t3 ]B = 6 3 7 = 2~e1 + 3~e2 ~e4 .
[1]B = 6
405
405
405
1
0
0
(b) What is dim P3 ?
(c) Compute [t2 ]B , [t3 ]B , and [3
t + 2t2 + 5t3 ]B .
Using the coordinate mapping, we can view T as a linear transformation R4
P3
[ ]B
✏
R
T
/ R2
>
S
/
R2 :
.
S
4
S should act the same way on a vector of coordinates as T acts on the polynomial that has
those B-coordinates, namely S([~p]B ) = T (~p). Since the domain and codomain of S are vector
spaces of the form Rn , we can find a matrix A such that S(~x) = A~x for all ~x in R4 . In fact,
the columns of A are S(~e1 ), S(~e2 ), S(~e3 ), S(~e4 ). Let’s compute these vectors. Here are the
first two:


1
1
S(~e1 ) = S([1]B ) = T (1) =
,
S(~e2 ) = S([t]B ) = T (t) =
.
1
2
(d) Compute S(~e3 ) and S(~e4 ).
(e) Write down the matrix A.
(f) Use A and your answer in (c) to compute S([3
your answer matches (a).
2
t + 2t2 + 5t3 ]B ). Check to make sure
You now know how S acts, so you can compute the kernel of S, which equals Nul A, and
the range of S, which equals Col A. (Recall that the kernel of S is the set of all vectors ~x
such that S(~x) = ~0 and the range of S is the set of all outputs S(~x) for all ~x.)
(g) Compute a basis for Nul A by row reducing A.
(h) Compute a basis for Col A. (Hint: use the pivot columns of A.)
(i) Compute the dimensions of the kernel of S and the range of S.
(j) Is S one-to-one? Is S onto?
Now that you have mastered S, you can use the fact that the coordinate mapping is an
isomorphism to deduce the same information about T .
(k) Compute a basis for the kernel of T using your basis for the kernel of S. Check your
vectors are really in the kernel of T by applying T to them.
3
(l) Without doing any work, why is the range of T equal to the range of S?
(m) Compute the dimensions of the kernel of T and the range of T .
(n) Is T one-to-one? Is T onto?
Now that you have deduced a lot of information about T , you should check whether your
results make sense by returning to the definition of T at the beginning.
(o) The fact that T is onto means that for any real numbers a and b, there is a polynomial
p~ in P3 such that p~(1) = a andp~(2) = b. Is this surprising? Compute such a polynomial
a
by finding a solution to A~x =
using row reduction.
b
(p) Factor the polynomials in your basis for the kernel of T in (k). (Feel free to use
www.wolframalpha.com to help you.) Why are these factorizations not surprising given
the way we defined T ?
4