6.
Linear Transformations
Let V, W be vector spaces over a field F. A
function that maps V into W , T : V → W ,
is called a linear transformation from V
to W if for all vectors u and v in V and all
scalars c ∈ F
(a) T (u + v) = T (u) + T (v)
(b) T (cu) = cT (u)
Basic Properties of Linear Transformations
Let T : V → W be a function.
(a) If T is linear, then T (0) = 0
(b) T is linear if and only if T (av + w) =
aT (v)+T (w) for all v, w in V and a ∈ F.
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In the special case where V = W , the linear
transformation T : V → V is called a linear
operator on V .
Examples
1. T : R2 → R2 s.t. T (a, b) = (2a + b, a)
2. T : Mn(R) → Mn(R) s.t. T (A) = AT
3. T : Pn(R) → Pn−1(R) s.t.
T (f (x)) = f 0(x)
4. C(R) is the space of cts real valued
functions on R. Fix a, b ∈ R s.t. a < b.
Then
Z b
T : C(R) → R s.t. T (f ) = f (t) dt.
a
5. Identity operator: For any V ,
I : V → V s.t. I(x) = x
6. Zero transformation: For any V, W ,
T0 : V → W s.t. T0(x) = 0
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Kernel and Image
Definitions
Let T : V → W be a linear transformation.
The set of vectors in V that T maps into 0
is called the kernel of T . It is denoted by
ker(T ). In mathematical notation:
ker(T ) = {v ∈ V | T (v) = 0}
The set of all vectors in W that are images
under T of at least one vector in V is called
the Image of T ; it is denoted by Im(T ). In
mathematical notation:
Im(T ) = {w ∈ W |w = T (v) for some v ∈ V }
Theorem
Let T : V → W be linear. Then ker(T ) and
Im(T ) are subspaces of V and W respec-
tively.
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Example
T : R3 → R2 s.t. T (a, b, c) = (a − b, 2c)
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Theorem
If T : V → W is a linear transformation and
{v1, v2, . . . , vn} forms a basis for V , then
Im(T ) = span(T (v1), T (v2), . . . , T (vn))
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Rank and Nullity
Definitons If T : U → V is a linear trans-
formation,
• the dimension of the image of T is called
the rank of T and is denoted by rank(T ),
• the dimension of the kernel is called the
nullity of T and is denoted by nullity(T ).
Example
Let U be a vector space of dimension n,
with basis {u1, u2, . . . , un}, and let T : U →
U be a linear operator defined by
T (ui) = ui+1, i = 1, . . . , n − 1, T (un) = 0
Find bases for ker(T ) and Im(T ) and deter-
mine rank(T ) and nullity(T ).
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Theorem
If T : V → W is a linear transformation
from an n-dimensional vector space V to a
vector space W , then
rank(T ) + nullity(T ) = dim(V ) = n
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Theorem
Let T : V → W be linear. Then T is injec-
tive if and only if ker(T ) = {0}.
Theorem
Let T : V → W be linear and dim(V ) =
dim(W ). Then the following are equiva-
lent:
• T is injective
• T is surjective
• rank(T ) = dim(V )
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Theorem
Suppose that {v1, v2, . . . , vn} is a basis for V.
For w1, w2, . . . , wn in W there exists exactly
one linear transformation T : V → W such
that T (vi) = wi, i = 1, 2, . . . , n.
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Corollary
Let {v1, v2, . . . , vn} be a basis for V and let
T1, T2 : V → W be linear s.t. T1(vi) =
T2(vi) for i = 1, 2, . . . , n. Then T1 = T2.
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Example
Let T : R3 → R2 s.t. T (a, b, c) = (a − b, 2c).
Suppose U : R3 → R2 is linear and
U (1, 1, 1) = (0, 2), U (1, 0, −1) = (1, −2),
U (0, −1, 1) = (1, −2).
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Quiz
True or false?
• If T (x + y) = T (x) + T (y) then T is
linear.
• If T : V → W is linear then T (0V ) = 0W .
• T is injective if and only if the only vec-
tor x satisfying T (x) = 0 is x = 0.
• Given x1, x2 ∈ V and y1, y2 ∈ W , there
exists a linear transformation T : V →
W s.t. T (x1) = y1 and T (x2) = y2.
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