3.1 Image and Kernal of a Linear Trans- Definition. Image

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3.1 Image and Kernal of a Linear Trans-
formation

Image

Definition.
The image of a function consists of all the
values the function takes in its codomain. If f
is a function from X to Y , then

image(f) = {f (x): x ∈ X}

= {y ∈ Y : y = f (x), for some x ∈ X}

Example. See Figure 1.

Example. The image of

f (x) = ex

consists of all positive numbers.

Example.

b ∈ im(f ), c (cid:54)∈ im(f ) See Figure 2.

Example. f (t) =

(See Figure 3.)

(cid:34)

(cid:35)

cos(t)
sin(t)

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If the function from X to Y is in-
Example.
vertible, then image(f ) = Y . For each y in Y ,
there is one (and only one) x in X such that
y = f (x), namely, x = f −1(y).

Example. Consider the linear transformation
T from R3 to R3 that projects a vector or-
thogonally into the x1 − x2-plane, as illustrate
in Figure 4. The image of T is the x1−x2-plane
in R3.

Example. Describe the image of the linear
transformation T from R2 to R2 given by the
matrix

(cid:34)

(cid:35)

1 3
2 6

A =

Solution

(cid:34)

(cid:35)

(cid:34)

(cid:35)

(cid:34)

(cid:35) (cid:34)

(cid:35)

T

= A

=

x1
x2

1 3
2 6

x1
x2

x1
x2

2

(cid:34)

(cid:35)

(cid:34)

(cid:35)

(cid:34)

(cid:35)

(cid:34)

(cid:35)

= x1

+ x2

= x1

+ 3×2

1
2

1
2

1
2

3
6

(cid:35)

(cid:34)

1
2

= (x1 + 3×2)

See Figure 5.

Example. Describe the image of the linear
transformation T from R2 to R3 given by the
matrix

A =











1 1
1 2
1 3

Solution

(cid:34)

(cid:35)

T

x1
x2






=

(cid:34)






(cid:35)

x1
x2

1 1
1 2
1 3

= x1


 + x2








1
1
1











1
2
3

See Figure 6.

Definition. Consider the vectors (cid:126)v1, (cid:126)v2, . . . ,
(cid:126)vn in Rm. The set of all linear combinations of
the vectors (cid:126)v1, (cid:126)v2, . . . , (cid:126)vn is called their span:

span((cid:126)v1, (cid:126)v2, . . . , (cid:126)vn)
={c1(cid:126)v1 + c2(cid:126)v2 + . . . + cn(cid:126)vn: ci arbitrary scalars}

Fact The image of a linear transformation

T ((cid:126)x) = A(cid:126)x

is the span of the columns of A. We denote
the image of T by im(T ) or im(A).

Justification

T ((cid:126)x) = A(cid:126)x =






|
|
(cid:126)v1 . . . (cid:126)vn
|
|




















x1
x2
…
xn

= x1 (cid:126)v1 + x2 (cid:126)v2 + . . . + xn (cid:126)vn.

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Fact: Properties of the image

(a). The zero vector is contained in im(T ),
i.e. (cid:126)0 ∈ im(T ).

(b). The image is closed under addition:
If (cid:126)v1, (cid:126)v2 ∈ im(T ), then (cid:126)v1 + (cid:126)v2 ∈ im(T ).

(c). The image is closed under scalar multipli-
cation: If (cid:126)v ∈ im(T ), then k(cid:126)v ∈ im(T ).

Verification

(a). (cid:126)0 ∈ Rm since A(cid:126)0 = (cid:126)0.

(b). Since (cid:126)v1 and (cid:126)v2 ∈ im(T ), ∃ (cid:126)w1 and (cid:126)w2 st.
T ( (cid:126)w1) = (cid:126)v1 and T ( (cid:126)w2) = (cid:126)v2. Then, (cid:126)v1 + (cid:126)v2 =
T ( (cid:126)w1) + T ( (cid:126)w2) = T ( (cid:126)w1 + (cid:126)w2), so that (cid:126)v1 + (cid:126)v2
is in the image as well.

(c). ∃ (cid:126)w st. T ( (cid:126)w) = (cid:126)v. Then k(cid:126)v = kT ( (cid:126)w) =
T (k (cid:126)w), so k(cid:126)v is in the image.

4

Example. Consider an n × n matrix A. Show
that im(A2) is contained in im(A).

Hint: To show (cid:126)w is also in im(A), we need to
find some vector (cid:126)u st. (cid:126)w = A(cid:126)u.

Solution

Consider a vector (cid:126)w in im(A2). There exists
a vector (cid:126)v st. (cid:126)w = A2(cid:126)v = AA(cid:126)v = A(cid:126)u where
(cid:126)u = A(cid:126)v.

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Definition. Kernel

The kernel of a linear transformation T ((cid:126)x) =
A(cid:126)x is the set of all zeros of the transformation
(i.e., the solutions of the equation A(cid:126)x = (cid:126)0. See
Figure 9.

We denote the kernel of T by ker(T ) or ker(A).

For a linear transformation T from Rn to Rm,

• im(T ) is a subset of the codomain Rm of

T , and

• ker(T ) is a subset of the domain Rn of T .

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Example. Consider the orthogonal project onto
the x1 − x2−plane, a linear transformation T
from R3 to R3. See Figure 10.

The kernel of T consists of all vectors whose
orthogonal projection is (cid:126)0. These are the vec-
tors on the x3−axis (the scalar multiples of (cid:126)e3).

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