A First Course in Linear Algebra, 2017a

9.8. The Kernel And Image Of A Linear Map 513
Example 9.79: Kernel and Image of a Transformation
Let T : P 1 → R be the linear transformation defined by
T (p(x)) = p(1) for all p(x) ∈ P 1 .
Find the kernel and image of T .
Solution. We will first find the kernel of T . It consists of all polynomials in P 1 that have 1 for a root.
ker(T) = {p(x) ∈ P 1 | p(1)=0}
= {ax + b | a,b ∈ R and a + b = 0}
= {ax − a | a ∈ R}
Therefore a basis for ker(T ) is
{x − 1}
Notice that this is a subspace of P 1 .
Now consider the image. It consists of all numbers which can be obtained by evaluating all polynomials
in P 1 at 1.
im(T ) = {p(1) | p(x) ∈ P 1 }
= {a + b | ax + b ∈ P 1 }
= {a + b | a,b ∈ R}
= R
Therefore a basis for im(T ) is
{1}
Notice that this is a subspace of R, and in fact is the space R itself.
♠
Example 9.80: Kernel and Image of a Linear Transformation
Let T : M 22 ↦→ R 2 be defined by
[
a b
T
c d
]
=
[
a − b
c + d
]
Then T is a linear transformation. Find a basis for ker(T ) and im(T).
Solution. You can verify that T represents a linear transformation.
Now we want [ to find ] a way to describe all matrices A such that T (A)=⃗0, that is the matrices in ker(T).
a b
Suppose A = is such a matrix. Then
c d
[ ] [ ] [ ]
a b a − b 0
T = =
c d c + d 0