Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Chapter 4: Linear Transformations
Paul Pearson
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Functions
Definition
Let X and Y be sets.
1. A function f : X → Y is a rule that assigns exactly one
element f (x) ∈ Y to each element x ∈ X.
2. The domain of f is the set X. We write dom(f ) = X.
3. The codomain of f is the set Y. We write codom(f ) = Y.
4. The image of f is the set of all outputs of f :
im(f ) = {y ∈ Y | y = f (x) for some x ∈ X}
= {f (x) ∈ Y | x ∈ X}
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Properties of functions
Definition
Let X and Y be sets, and let f : X → Y be a function.
1. f is surjective if every element in the codomain has an
element in the domain that maps to it:
a. If for every y ∈ Y there exists an x ∈ X such that y = f (x), or
b. im(f ) = Y
2. f is injective if every pair of distinct inputs get mapped to
distinct outputs:
a. If for all x1 6= x2 in X, f (x1 ) 6= f (x2 ) in Y, or equivalently
b. whenever f (x1 ) = f (x2 ) in Y, x1 = x2 in X.
3. f is bijective if it is both injective and surjective.
Example
1. f : R → R given by f (x) = x2
2. g : R → R given by g(x) = 2x + 1
3. h : R → R given by h(x) = 2x
4. k : R → R given by k(x) = x3 − x
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Properties of functions
Definition
Let X, Y and Z be sets, and f : X → Y and g : Y → Z be
functions.
1. The identity function idX : X → X is defined by idX (x) = x
for all x ∈ X.
2. The composition g ◦ f : X → Z is defined by
(g ◦ f )(x) = g(f (x)) for all x ∈ X.
3. If there exists a function f −1 : Y → X such that
f −1 ◦ f = idX and f ◦ f −1 = idY
we say f is invertible and f −1 is an inverse function for f .
Theorem
A function is invertible if and only if it is bijective.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Definition
Let f : V → W be a function from a vector space V to a vector
space W. We say that f is a linear transformation if it respects the
operations of vector addition and scalar multiplication:
1. For all v1 , v2 ∈ V, f (v1 + v2 ) = f (v1 ) + f (v2 ), and
2. For all v ∈ V and all α ∈ R, f (αv) = αf (v).
Example
Domain = R2
y
Codomain = R2
y
f (v1 + v2 ) = f (v1 ) + f (v2 )
v1 + v2
v2
f (v1 )
v1
x
f (v2 )
f
x
−1.5v1
f (−1.5v1 ) = −1.5f (v1 )
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Example
Is f (x) = x2 a linear transformation? Identify the domain,
codomain, image, and whether the function is injective,
surjective, or bijective.
Not linear: f (x1 + x2 ) = (x1 + x2 )2 6= x21 + x22 = f (x1 ) + f (x2 )
Not injective: f (1) = f (−1) = 1
Not surjective: im(f ) = {x2 | x ∈ R} = {y | y ≥ 0} =
6 R
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Which of the following functions are linear transformations?
Identify the domain, codomain, image, and whether the
function is injective, surjective, or bijective.
1. f (x) = 2x + 1
2. g(x) = 3x
3. h(x) = sin(x)
All have domain = R and codomain = R. Only g is linear. f and
g are bijective, and h is neither injective nor surjective.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Which of the following functions are linear transformations?
Identify the domain, codomain, image, and whether the
function is injective, surjective, or bijective. If possible, write
the formula for the function using matrices.
1. f (x, y) = 2x + 3y − 1
2. g(x, y) = 3x − 2y
g
hx, yi
y
3x − 2y
3
2
g(e1 )
e2
1
x
e1
0
−1
g(e2 )
−2
Domain = R2
−3
Codomain = R
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Which of the following functions are linear transformations?
Identify the domain, codomain, image, and whether the
function is injective, surjective, or bijective. If possible, write
the formula for the function using matrices.
1. f (x) = hx, 3x + 1i
2. g(x) = h2x, 3xi
g
3
h2x, 3xi
y
x
2
g(e1 )
1
e1
x
0
−1
im(g)
−2
−3
Codomain = R2
Domain = R
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Which of the following functions are linear transformations?
Identify the domain, codomain, image, and whether the
function is injective, surjective, or bijective. If possible, write
the formula for the function using matrices.
1. f (x, y) = hy − 2x, 2y − xi
2. g(x, y) = h3x + y − 2, x − 2yi
f
hx, yi
y
h−2x + y, −x + 2yi
y
e2
f (e2 )
x
x
e1
f (e1 )
im(f )
Domain = R2
Codomain = R2
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Which of the following functions are linear transformations?
Identify the domain, codomain, image, and whether the
function is injective, surjective, or bijective. If possible, write
the formula for the function using matrices.
1. D : P → P given by D(f ) =
df
dt
= f 0 (t)
2. L : P → P given by
L(f ) = (D2 − D + 3)(f ) = D(D(f )) − D(f ) + 3f
3. t : Mm,n (R) → Mn,m (R) given by t(A) = AT
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: dilations
Dilation by a factor of 2 in x-direction
y
v
e2
x
e1
x
!
f (e1 )
Dilation by a factor of 3 in y-direction
y
y
g(e2 )
v
e2
x
e1
f (v)
f (e2 )
f
2 0
0 1
y
g(v)
g
1 0
0 3
x
!
g(e1 )
Dilation by a factor of 2 in x-dir. and 3 in y-dir.
y
y
h(e2 )
e2
x
e1
v
h
2 0
0 3
!
x
h(e1 )
h(v)
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: shears
Shear in x-direction
y
y
f (v)
f (e2 )
e2
v
e1
x
f
1 2
0 1
x
!
f (e1 )
Shear in y-direction
y
v
y
g(e2 )
e2
e1
x
g
1 0
1 1
!
g(e1 )
x
g(v)
Remark
Dilations and shears naturally arise from row operations:
I
I
Dilations arise from αRi → Ri
Shears arise from Ri + αRj → Ri
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: reflections
Reflection across y-axis
y
e2
y
f (v)
v
x
e1
f
−1 0
0 1
f (e2 )
!
x
f (e1 )
Reflection across x-axis
y
e2
y
v
x
e1
g
1
0
0 −1
g(e1 )
!
x
g(e2 )
g(v)
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: reflections
Reflection across y = x
y
y
v e2
f (e1 )
e1
x
f
0 1
1 0
x
f (e2 )
!
f (v)
Reflection across y = −x
y
v e2
e1
x
g(v)
g
0 −1
−1
0
y
x
g(e2 )
!
g(e1 )
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: rotations
Rotation CCW by θ
y
y
e2
f (e1 )
e1
x
f
f (e2 )
cos θ − sin θ
sin θ
cos θ
v
x
!
f (v)
Reflection thru origin = rotation by 180◦
y
y
v
e2
e1
x
g
−1
0
0 −1
!
g(v)
g(e1 )
g(e2 )
x
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: projections
Projection onto x-axis
y
v
y
e2
e1
x
f
f (v)
1 0
0 0
!
x
f (e1 )
Projection onto y-axis
y
v
y
g(v)
e2
e1
x
g
g(e2 )
0 0
0 1
!
x
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Geometry of linear transformations: projections
Definition
A linear transformation f : Rn → Rn given by f (x) = Px for
some P ∈ Mn,n (R) is a projection if and only if P2 = P.
T
All rank one projection matrices P have the form bb
for some
bT b
n
column vector b ∈ R . Projection matrices of higher rank are
sums of rank one projection matrices.
Projection onto b
a
y
y
b
Pb
x
f
P =
b=
2
1
!
implies P =
x
bbT
bT b
bbT
=
bT b
Pa
0.8 0.4
0.4 0.2
!
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Kernel and image are subspaces
Definition
Let f : V → W be a linear transformation, and let 0W be the zero
vector in W. The kernel of f is
ker(f ) = {v ∈ V | f (v) = 0W .}
Example
Find the kernel, image, their dimensions, and the sum of their
dimensions for the following linear transformation.
z
z
im(f )
y
x
ker(f )
Domain = V = R3
f








1 −1 0 


0 0 0


0 0 2
y
x
Codomain = W = R3
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Theorem
Let f : V → W be a linear transformation.
1. f (0V ) = 0W .
2. ker(f ) is a subspace of V.
3. im(f ) is a subspace of W.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Properties of linear transformations
Theorem
Let f : V → W be a linear transformation.
1. f is injective if and only if ker(f ) = {0}.
2. If the span of B is V, then the span of f (B) is im(f ).
3. If f is injective and B is a linearly independent subset of V, then
f (B) is a linearly independent subset of W.
I
In particular, if f is injective and B is a basis of V, then f (B) is a
basis of im(f ).
4. If f is injective, then rank(f ) = dim(V).
5. If dim(V) < ∞, then dim(V) = rank(f ) + null(f ).
Theorem
The composition of linear transformations is a linear transformation.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Find the kernel, image, their dimensions, and the sum of their
dimensions for the following linear transformations.
1. f (x) = 5x
2. g(x, y) = 3x − 2y
g
hx, yi
y
ker(g)
3x − 2y
im(g)
x
Domain = R2
Codomain = R
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Find the kernel, image, their dimensions, and the sum of their
dimensions for the following linear transformations.
1. f (x) = h2x, 3xi
2. g(x, y) = h2x − 3y, −8x + 12yi
hx, yi
y
g
h2x − 3y, 6y − 4xi
y
ker(g)
x
x
im(g)
Domain = R2
Codomain = R2
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations
Exercise
Find the kernel, image, their dimensions, and the sum of their
dimensions for the following linear transformations.
df
0
dt = f (t)
d2 f
= dt2 = f 00 (t)
1. D : P4 → P4 given by D(f ) =
2. D2 : P4 → P4 given by D(f )
3. t : M2,2 (R) → M2,2 (R) given by t(A) = A − AT
Definition
Let f : V → W be a linear transformation.
1. null(f ) = dim(ker(f ))
2. rank(f ) = dim(im(f ))
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Isomorphisms
Theorem
Let f : V → W be a bijective linear transformation. Then the inverse
function f −1 : W → V is a linear transformation.
Definition
Let f : V → W be a bijective linear transformation. We say that f
is an isomorphism between V and W, and that V and W are
isomorphic. We write V ∼
= W.
Example
1. The coordinate vector transformation f : P2 → R3 defined
by f (a0 + a1 t + a2 t2 ) = ha0 , a1 , a2 i is an isomorphism
P2 ∼
= R3 .
2. Find an isomorphism M2,2 (R) ∼
= R4 .
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Isomorphisms
Theorem
Let V and W be finite dimensional vector spaces and f : V → W a
linear transformation.
1. If f is injective, then dim(V) ≤ dim(W).
2. If f is surjective, then dim(V) ≥ dim(W).
3. If f is an isomorphism, then dim(V) = dim(W).
4. If dim(V) = dim(W), then the following are equivalent:
a. f is injective.
b. f is surjective.
c. f is an isomorphism.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Isomorphisms
Theorem
Let V and W be vector spaces, and let B = {v1 , . . . , vn } be a basis of
V. Let w1 , . . . , wn be elements in W. Then there is a unique linear
transformation f : V → W such that
f (vi ) = wi
for all i. Moreover,
1. f is injective if and only if {w1 , . . . , wn } is linearly independent.
2. f is surjective if and only if span{w1 , . . . , wn } is equal to W.
3. f is an isomorphism if and only if {w1 , . . . , wn } is a basis of W.
Corollary
Let V and W be finite dimensional vector spaces. Then
V∼
= W ⇐⇒ dim(V) = dim(W).
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Injective, surjective, or an isomorphism?
If f is a linear transformation and f (vi ) = wi for all i, is f
injective, surjective, an isomorphism, or none of these?
y
y
w1
v1
x
f
x
v2
w2
Not collinear
Domain = R2
Collinear
Codomain = R2
y
y
w1
v2
w2
x
f
x
v1
Not collinear
Domain = R2
Not collinear
Codomain = R2
In R2 : basis ⇐⇒ not collinear ⇐⇒ Atriangle 6= 0 ⇐⇒ det 6= 0
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Injective, surjective, or an isomorphism?
If f is a linear transformation and f (vi ) = wi for all i, is f
injective, surjective, an isomorphism, or none of these?
z
z
v2
w2
v3
y
f
w3
v1
x
y
w1
x
Not coplanar
Domain = R3
Coplanar
Codomain = R3
z
z
w3
v3
v2
y
f
w2
w1
v1
x
y
x
Not coplanar
Domain = R3
Not coplanar
Codomain = R3
In R3 : basis ⇐⇒ not coplanar ⇐⇒ Vtet 6= 0 ⇐⇒ det 6= 0
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Injective, surjective, or an isomorphism?
If f is a linear transformation and f (vi ) = wi for all i, is f
injective, surjective, an isomorphism, or none of these?
y
z
v1
x
f
w1
w2
y
v2
x
Domain = R2
Codomain = R3
y
z
v1
w1
x
f
y
v2
w2
x
Domain = R2
Collinear
Codomain = R3
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Injective, surjective, or an isomorphism?
If f is a linear transformation and f (vi ) = wi for all i, is f
injective, surjective, an isomorphism, or none of these?
y
z
v2
w3
w1
f
v1
x
y
w2
v3
Not coplanar
x
Domain = R3
Codomain = R2
y
z
v2
w1
f
v1
x
w2
y
x
v3
Not coplanar
Domain = R3
w3
Codomain = R2
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Injective, surjective, or an isomorphism?
If possible, give examples of a linear transformation f that are
injective, surjective, an isomorphism, or none of these.
1. f : R2 → R3
2. f : R3 → R3
3. f : R4 → R3
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Isomorphisms
Theorem
Let f , h : V → W and g : W → X be linear transformations.
1. (Adding:) (f + h) : V → W given by (f + h)(v) = f (v) + h(v)
is a linear transformation.
2. (Scaling:) (αf ) : V → W given by (αf )(v) = αf (v) is a linear
transformation.
3. (Composition:) (g ◦ f ) : V → X given by (g ◦ f )(v) = g(f (v)) is
a linear transformation.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrices of linear transformations
Theorem
Every linear transformation f : Rk → Rn defines a matrix
A ∈ Mn,k (R), and every matrix A ∈ Mn,k (R) defines a linear
transformation f : Rk → Rn :
1. Let f : Rk → Rn be a linear transformation, and let
A = (f (e1 ) | · · · | f (ek )).
Then for all x ∈ Rk we have f (x) = Ax.
2. Let A ∈ Mn,k (R), and let f (x) = Ax. Then f : Rk → Rn is a
linear transformation.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear Transformations and Matrices
Exercise
Let f : R2 → R3 be a linear transformation, and suppose that
f (1, 2) = h1, 2, 3i,
f (3, 1) = h1, −1, 2i.
z
y
e2
x1
f (x2 ) = h1, −1, 2i
x2
f
x
e1
f (x1 ) = h1, 2, 3i
f (e2 )
f (e1 )
A = (f (e1 ) | f (e2 ))
y
x
Domain = R
Codomain = R3
2
1. If f (x) = Ax, how many rows and columns must A have?
2. What is the matrix A that represents f ? Hint: since
f (x) = Ax,
(f (x1 ) | f (x2 )) = (Ax1 | Ax2 ) = A(x1 | x2 ) = AX.
3. What are f (e1 ) and f (e2 )?
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrices of linear transformations
Theorem
Let A ∈ Mm,n (R) and B ∈ Mk,m (R), and let f : Rn → Rm be the
linear transformation defined by f (x) = Ax and g : Rm → Rk be the
linear transformation defined by g(y) = By. Then g ◦ f : Rn → Rk is
the linear transformation defined by
(g ◦ f )(x) = BAx.
Theorem
Let A ∈ Mn,n (R), and let f : Rn → Rn be the linear transformation
defined by f (x) = Ax. f is an isomorphism if and only if A is an
invertible matrix, and, in that case, f −1 (x) = A−1 x.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Coordinate vectors
Definition
If B is a basis for a vector space V, and the order in which the
elements of B is taken into account, then we say B is an ordered
basis for V.
Remark
If B = {b1 , . . . , bk } is an ordered basis for a vector space V,
then any element v ∈ V can be written uniquely as a linear
combination
v = x1 b1 + · · · + xk bk .
Definition
Let B = {b1 , . . . , bk } be an ordered basis for a vector space V,
and let v = x1 b1 + · · · + xk bk . The coordinate representation of v
with respect to the ordered basis B is
[v]B = hx1 , . . . , xk iB ∈ Rk .
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Coordinate vectors
Standard basis S = {e1 , e2 }
y
b2
e2
b1
e1
x
[b1 ]S = 1e1 + 1e2 = h1, 1iS
[b2 ]S = −1e1 + 1e2 = h−1, 1iS
y
b2
b1
x
[b1 ]B = 1b1 + 0b2 = h1, 0iB
[b2 ]B = 0b1 + 1b2 = h0, 1iB
Custom basis B = {b1 , b2 }
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Coordinate vectors
Standard basis S = {e1 , e2 }
y
e2
−e1
−3e1
e1
x
[x]S = −1e1 − 3e2 = h−1, −3iS
x
y
b2
b1
x
[x]B = −2b1 − 1b2 = h−2, −1iB
−2b1
x
−b2
Custom basis B = {b1 , b2 }
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Coordinate vectors
Example
Let b1 = h1, 1iS , b2 = h3, −1iS , and B = {b1 , b2 }.
1. [b2 ]B = h0, 1iB means h3, −1iS = h0, 1iB . The book writes
this as [h3, −1i]B = h0, 1i.
2. What are the B-coordinates for h2, 0i? [h2, 0i]B = h1/2, 1/2i.
3. What is [h3, 2i]B ? h2.25, 0.25iB
y
b1
x
b2
Custom basis B = {b1 , b2 }
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Coordinate vectors
Exercise
Let B = {1 + t, 1 + t2 , t + t2 } be an ordered basis for P2 . Find:
1. [1 + 2t + 3t2 ]B
2. [2]B
Remark
Recall that a matrix A ∈ Mn,k (R) uniquely determines a linear
transformation f : Rk → Rn defined by f (x) = Ax. Further,
column i of the matrix A is f (ei ), so
A = (f (e1 ) | · · · | f (ek )).
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis matrix
Standard basis S = {e1 , e2 }
y
b1
b2
e2
x
e1
[b1 ]S =
1
2
!
S
[b2 ]S =
!
−1
1
S
xS = BxB =
xS
id
B=
1 −1
2
1
!
id
id
id
y
b1
b2
x
xB
Custom basis B = {b1 , b2 }
[b1 ]B =
1
0
!
B
[b2 ]B =
0
1
!
B
xB
1 −1
2
1
!
xB
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of coordinates (basis)
Suppose B = {b1 , . . . , bn } is a basis for Rn . Let S = {e1 , . . . , en }
be the standard basis for Rn . Suppose x is any arrow in Rn .
I (S-coordinates:) relative to the standard “unit square” grid
system, x has coordinate vector
xS = x1 e1 + · · · + xn en = hx1 , . . . , xn iS .
I (B-coordinates:) relative to the custom “parallelogram”
grid system, x has coordinate vector
xB = y1 b1 + · · · + yn bn = hy1 , . . . , yn iB . (Usually, xi 6= yi .)
I The identity transformation id : (Rn , B) → (Rn , S) does not
change the vector x (i.e., xB and xS are the same arrow!),
but it does change the coordinate vector description of x
from B-coordinates to S-coordinates. The identity
transformation xB 7→ xS is given by id(xB ) = BxB = xS ,
where the change of basis matrix is
B = ([b1 ]S | · · · | [bn ]S ).
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis matrix
Standard basis S = {e1 , e2 }
y
e2
1. What is the custom basis B?
b1
x
e1
2. What is the change of basis matrix from
B-coordinates to standard coordinates?
3. What is the change of basis matrix from
standard coordinates to B-coordinates?
b2
4. Define the change of coordinates
transformation id. What are id(h1, 0iB )
and id(h0, 1iB )?
id
y
b1
x
b2
Custom basis B = {b1 , b2 }
5. What is h1, −2iB in standard
coordinates? (Draw a picture to verify
your answer.)
6. What is h−3, 2iS in B-coordinates?
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Fundamental isomorphism theorem
Theorem
Any n-dimensional vector space V is isomorphic to Rn .
Remark
Because of this isomorphism, any question about an
n-dimensional vector space V can be translated into a question
in Rn .
Exercise
Let
S = {1 + t2 , t + t3 , t2 + t3 , 1 + t3 },
T = {1 + t2 , 1 − t − t2 , t + 2t2 , 2 + 3t − t2 }.
1. Is 1 + 3t a linear combination of the elements in S?
2. Is S a basis for P3 ?
3. Find a basis for span(T).
4. Is T linearly independent?
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations relative to a custom basis
Suppose f (xS ) = 40 68 xS , relative to the standard basis in the
domain and codomain. What is the matrix of f relative to the
basis B in the domain and C in the codomain?
Standard Basis S
xS
b2 e2 b1
e1
f
A
Standard Basis S
c2
e2
c1
e1
f (xS )
B id
xB
b2
C −1 id
b1
id C
c2
[f ]CB
c1
C −1 AB
[f ]CB (xB )
Custom basis B
Custom basis C
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Linear transformations relative to a custom basis
Suppose f : (Rk , S) → (Rn , S) is a linear transformation relative
to the standard basis in the domain and the codomain. Then
there is some matrix A ∈ Mn,k (R) so that f (xS ) = AxS for all xS
in the domain.
The linear transformation f relative to the basis B in the domain
and the basis C in the codomain is the function
[f ]CB : (Rk , B) → (Rn , C)
defined by
[f ]CB (xB ) = C−1 ABxB
where the change of basis matrices B and C have standard
coordinate vectors in their columns:
B = ([b1 ]S | · · · | [bk ]S ) and C = ([c1 ]S | · · · | [cn ]S ).
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis
Find the change of basis matrix from B-coordinates to
C-coordinates.
Standard Basis S
b2
e2
e1
Standard Basis S
id
c2 e2 c1
e1
I
b1
C −1 id
B id
b2
[id]CB
c2
id C
c1
C −1 B
b1
Custom basis B
Custom basis C
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis
We know how to change from a custom basis B to the standard
basis S for Rn . How can we change from a custom basis B to
another custom basis C for Rn ?
Take f to be the identity transformation
id : (Rn , S) → (Rn , S)
defined by id(xS ) = IxS = xS . The change of basis xB 7→ xC is
the linear transformation
[id]CB : (Rn , B) → (Rn , C)
defined by
[id]CB (xB ) = C−1 BxB = xC .
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis
Consider the vector space V of discrete time signals with 4
samples at t = 0, 0.25, 0.5, 0.75.
1. Show that the Haar wavelets and scaling functions
H = {h1 = h−1, 1, 0, 0i, h2 = h0, 0, −1, 1i, h3 = h1, 1, 0, 0i,
h4 = h0, 0, 1, 1i} form an orthogonal basis for V.
2. Find the change of basis matrix from H to the standard
basis.
3. Use the inverse of this change of basis matrix to write the
time signal s = h1, 4, 2, −4iS in terms of the basis H.
Custom Basis H
h1 =
h2 =
h3 =
h4 =
Standard Basis S
-1
1
0
0
0
0
-1
1
1
1
0
0
0
1
1
1
0
0
0
e2 =
0
1
0
0
e3 =
0
0
1
0
e4 =
0
0
0
1
id
H
id
H −1
0
e1 =
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Change of basis
1. Show that the Legendre polynomials
L = {1, t,
1
2 (−1
+ 3t2 ),
1
2 (−3t
+ 5t3 )}
form a basis for P3 .
2. Show that the Chebyshev polynomials of the first kind
C = {1, t, −1 + 2t2 , −3t + 4t3 }
form a basis for P3 .
3. Find the change of basis matrix from the Legendre
polynomials to the Chebyshev polynomials.
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrix of a linear transformation
Example
Let f : (R3 , S) → (R2 , S) be the linear transformation defined by
f (xS ) = AxS , where
1 −4
2
A=
.
3
2 −1
Let xS = h2, −1, 3iS and
B = {h1, 0, 1i, h1, −1, 1i, h1, 1, 0i},
C = {h3, 1i, h−2, −1i}.
1. Find f (xS ).
2. Find the matrix for [f ]CB .
3. Find xB and [f ]CB (xB ).
4. Find [f (xS )]C and show that [f ]CB (xB ) = [f (xS )]C .
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrix of a linear transformation
Exercise
Let f : (R2 , S) → (R2 , S) be the linear transformation defined by
f (x) = Ax, where
3 4
A=
−1 2
Let
B = {h3, 2i, h−1, 2i}, C = {h4, 1i, h3, −1i}.
and let xS = h2, 1iS .
1. Find f (xS ).
2. Find the matrix for [f ]CB .
3. Find xB and [f ]CB (xB ).
4. Find [f (xS )]C and show that [f ]CB (xB ) = [f (xS )]C .
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrix of a linear transformation
Exercise
Let f : P2 → P2 be the linear transformation defined by
f (p) = 2p − 3p0 + p00
Find [f ]CB for each of the following choices of B and C.
1. B = C = {1, t, t2 }
2. B = {1, 1 + t, 1 + t + t2 } and C = {1, t, t2 }
3. B = {1, t, t2 } and C = {1, 1 + t, 1 + t + t2 }
Linear Transformations Kernel, Image, and Isomorphism Change of basis Coordinate Vectors Custom Linear Transformations
Matrix of a linear transformation
Exercise
Let f : M2 (R) → M2 (R) be the linear transformation defined by
f (A) = A + AT .
1. Find [f ]SS , where S is the standard basis for M2 (R).
2. Find a basis for ker(f ).
3. Find a basis for im(f ).