1. a linear transformation from V into V
a group
a vector space V
a linear transformation from V into V
a review of the whole course
in one example
Pamela Leutwyler

2. G is a commutative group.
& a b c d e f g h i

3. & a b c d e f g h i G is a commutative group.
Note that the table has closure.
x&y is a member of G for ALL x and y.
ie: every entry on the table
belongs to G.
& a b c d e f g h i

4. & a b c d e f g h i G is a commutative group. i is the identity.
ix = x for ALL x
and xi = x for ALL x.
& a b c d e f g h i

5. & a b c d e f g h i G is a commutative group. i is the identity.
a is the inverse of f .
& a b c d e f g h i
af = I and fa = i

6. & a b c d e f g h i G is a commutative group. i is the identity.
a is the inverse of f .
b is the inverse of c .
d is the inverse of g .
h is the inverse of e .

7. & a b c d e f g h i G is a commutative group. (b&h)&d b&(h&d)
Here is one example
demonstrating associativity.
Remember:
(x&y)&z = x&(y&z)
for ALL z,y,z in G
& a b c d e f g h i
(b&h)&d
b&(h&d)

8. & a b c d e f g h i G is a commutative group. (b&h)&d b&(h&d) (a)&d
Here is one example
demonstrating associativity.
Remember:
(x&y)&z = x&(y&z)
for ALL z,y,z in G
& a b c d e f g h i
(b&h)&d
b&(h&d)
(a)&d
b&(b)

9. & a b c d e f g h i G is a commutative group. c c (b&h)&d b&(h&d)
Here is one example
demonstrating associativity.
Remember:
(x&y)&z = x&(y&z)
for ALL z,y,z in G
& a b c d e f g h i
(b&h)&d
b&(h&d)
(a)&d
b&(b) c c

10. & a b c d e f g h i G is a commutative group. i is the identity.
a is the inverse of f .
b is the inverse of c .
d is the inverse of g .
h is the inverse of e .
H = { a, f, i }
is a subgroup.
Note that it has closure.

11. & a b c d e f g h i G is a commutative group. H = { a, f, i }
is a subgroup.
Note that it has closure.

12. & a f i G is a commutative group. H = { a, f, i } is a subgroup.
Note that it has closure.

13. & a b c d e f g h i G is a commutative group. H = { a, f, i }
is a subgroup.
& a b c d e f g h i

14. & a b c d e f g h i G is a commutative group. H = { a, f, i }
is a subgroup.
& a b c d e f g h i
add b to every member of H
to form a COSET of H.
{ a, f, i } b+

15. & a b c d e f g h i G is a commutative group. H = { a, f, i }
is a subgroup.
& a b c d e f g h i
add b to every member of H
to form a COSET of H.
{ a, f, i } b+
{ g, e, b }

16. & a b c d e f g h i G is a commutative group. H = { a, f, i }
is a subgroup.
& a b c d e f g h i
The cosets of H are:
{ g, e, b }
{ h, d, c }

17. G is a commutative group.
& a b c d e f g h i

18. & a b c d e f g h i G is a commutative group.
We can build a VECTOR SPACE using G as our set of vectors,
and arithmetic mod 3 as our field of scalars.
& a b c d e f g h i

19. & a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2

20. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:
2(g&b)
(2g)&(2b)

21. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:
2(g&b)
(2g)&(2b)
d & c
2  h

22. = let’s verify some vector space properties: 2(g&b) (2g)&(2b) d & ch i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties: =
2(g&b)
(2g)&(2b)
d & c
2  h e e

23. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:

24. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:
( )  d
( 2 d ) + ( 2  d )
1  d
g + g d d

25. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:

26. let’s verify some vector space properties:
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
let’s verify some vector space properties:
( 2 x 2 )  a
2  ( 2  a )
1  a
2  f a a

27. v we will name this vector space V & a b c d e f g h i . + 1 2 x a b c1 2 x . a b c d e f g h i 1 2
we will name this vector space V

28. v { a, b } is a basis for V 0a + 0b = i 1a + 2b = h
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
0a + 0b = i
1a + 2b = h
Examine all linear combinations
of a and b to the right.
Every vector in V is a linear
combination of a and b.
0a + 1b = b
2a + 0b = f
0a + 2b = c
2a + 1b = e
1a + 0b = a
2a + 2b = d
1a + 1b = g

29. v { a, b } is a basis for V 0a + 0b = i 1a + 2b = h
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
0a + 0b = i
1a + 2b = h
Examine all linear combinations
of a and b to the right.
Every vector in V is a linear
combination of a and b.
0a + 1b = b
2a + 0b = f
0a + 2b = c
2a + 1b = e
1a + 0b = a
2a + 2b = d
The only linear combination
to produce i is trivial.
1a + 1b = g

30. v { a, b } is a basis for V dim V = 2 {a,b} spans V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
dim V = 2
{a,b} spans V
{a,b} is independent
{ a, b } is a basis for V
0a + 0b = i
1a + 2b = h
Examine all linear combinations
of a and b to the right.
Every vector in V is a linear
combination of a and b.
0a + 1b = b
2a + 0b = f
0a + 2b = c
2a + 1b = e
1a + 0b = a
2a + 2b = d
The only linear combination
to produce i is trivial.
1a + 1b = g

31. v T is a linear transformation from V into V & a b c d e f g h i .+ 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
T(b)=e
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

32. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

33. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
the RANGE of T = { h, e, i }
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

34. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
0d = i d = d d = g
{ d } is a basis for the null space
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
the RANGE of T = { h, e, i }
T(f)=e
1a + 2b = h
0h = i h = h h = e
{ h } is a basis for the range
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

35. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
0d = i d = d d = g
{ d } is a basis for the null space
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
the RANGE of T = { h, e, i }
T(f)=e
1a + 2b = h
0h = i h = h h = e
{ h } is a basis for the range
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i
dim domain = dim null space + dim range

36. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
T(c)=h
Note that {d,g,i} is a subspace
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

37. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
0a + 0b = i
the NULL SPACE of T = { d, g, i }
T(b)=e
0a + 1b = b
T(c)=h
Note that {d,g,i} is a subspace (subgroup)
0a + 2b = c
T(d)=i
1a + 0b = a
Note the cosets of the null space.
Every element within a given coset
has the same image.
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

38. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

39. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
first column is T(a) = h = 1a + 2 b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

40. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
first column is T(a) = h = 1a + 2 b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
second column is T(b) = e = 2a + 1 b
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

41. v ( 1 2 ) T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
first column is T(a) = h = 1a + 2 b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
second column is T(b) = e = 2a + 1 b
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 )
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

42. v ( 1 2 ) ( 2 ) 1 T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e ( 2 ) 1
0a + 1b = b
T(c)=h
verify: e =
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 )
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

43. & a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e ( 2 ) 1
0a + 1b = b
T(c)=h
verify: e =
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 ) ( 2 ) 1 ( 1 ) 2
2a + 0b = f
T(g)=i = = h
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

44. v ( 1 2 ) T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 )
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

45. v ( 1 2 ) ( 2 ) ( 2 ) ( ) T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e ( 2 )
0a + 1b = b
T(c)=h
verify: d =
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 ) ( 2 ) ( )
2a + 0b = f
T(g)=i = = i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

46. v ( 1 2 ) T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e
0a + 1b = b
T(c)=h
Check some other cases.
This matrix times
the coordinates of a vector v
relative to {a,b} will give
the coordinates of
T v relative to {a,b} .
0a + 2b = c
T(d)=i
1a + 0b = a
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h ( 1 2 )
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

47. v T is a linear transformation from V into V
& a b c d e f g h i v + 1 2 x . a b c d e f g h i 1 2
{ a, b } is a basis for V
T is a linear transformation from V into V
T(a)=h
to find the MATRIX for T relative to the basis {a, b} :
0a + 0b = i
T(b)=e ( 1 2 )
0a + 1b = b
T(c)=h
0a + 2b = c
T(d)=i
1a + 0b = a
T is not the only linear transformation from V into V.
Since every linear transformation corresponds to a
2x2 matrix, you can create a different example by
choosing a different set of 4 scalars to build the
matrix. For example defines another linear
transformation.
T(e)=h
1a + 1b = g
T(f)=e
1a + 2b = h
2a + 0b = f
T(g)=i
2a + 1b = e
T(h)=e
2a + 2b = d
T(i)=i

48. V into V is called a LINEAR ALGEBRA.
& a b c d e f g h i + 1 2 x . a b c d e f g h i 1 2
The set of ALL linear transformations ( ie: 2x2 matrices of scalars) from
V into V is called a LINEAR ALGEBRA.
Since there are 3 scalars in this system, there are 3x3x3x3 = 81 ways to
form a 2x2 matrix and there are 81 different linear transformations from
V into V.