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
G is a commutative group. & a b c d e f g h i
& 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
& 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
& 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
& 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 .
& 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)
& 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)
& 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
& 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.
& 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.
& a f i G is a commutative group. H = { a, f, i } is a subgroup. Note that it has closure.
& 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
& 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+
& 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 }
& 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 }
G is a commutative group. & a b c d e f g h i
& 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
& 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: & 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) (2g)&(2b)
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) (2g)&(2b) d & c 2 h
= let’s verify some vector space properties: 2(g&b) (2g)&(2b) d & c 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) (2g)&(2b) d & c 2 h e e
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:
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 + 2 ) d ( 2 d ) + ( 2 d ) 1 d g + g d d
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:
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
v we will name this vector space V & a b c d e f g h i . + 1 2 x a b c 1 2 x . a b c d e f g h i 1 2 we will name this vector space V
v { a, b } is a basis for V 0a + 0b = i 1a + 2b = 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 0a + 0b = i 1a + 2b = h Examine all linear combinations of a and b to the right. Every vector in V is a linear combination of a and b. 0a + 1b = b 2a + 0b = f 0a + 2b = c 2a + 1b = e 1a + 0b = a 2a + 2b = d 1a + 1b = g
v { a, b } is a basis for V 0a + 0b = i 1a + 2b = 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 0a + 0b = i 1a + 2b = h Examine all linear combinations of a and b to the right. Every vector in V is a linear combination of a and b. 0a + 1b = b 2a + 0b = f 0a + 2b = c 2a + 1b = e 1a + 0b = a 2a + 2b = d The only linear combination to produce i is trivial. 1a + 1b = g
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 0a + 0b = i 1a + 2b = h Examine all linear combinations of a and b to the right. Every vector in V is a linear combination of a and b. 0a + 1b = b 2a + 0b = f 0a + 2b = c 2a + 1b = e 1a + 0b = a 2a + 2b = d The only linear combination to produce i is trivial. 1a + 1b = g
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 0a + 0b = i T(b)=e 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g the RANGE of T = { h, e, i } T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b 0d = i 1d = d 2d = g { d } is a basis for the null space T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g the RANGE of T = { h, e, i } T(f)=e 1a + 2b = h 0h = i 1h = h 2h = e { h } is a basis for the range 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b 0d = i 1d = d 2d = g { d } is a basis for the null space T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g the RANGE of T = { h, e, i } T(f)=e 1a + 2b = h 0h = i 1h = h 2h = e { h } is a basis for the range 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i dim domain = dim null space + dim range
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b T(c)=h Note that {d,g,i} is a subspace 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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 0a + 0b = i the NULL SPACE of T = { d, g, i } T(b)=e 0a + 1b = b T(c)=h Note that {d,g,i} is a subspace (subgroup) 0a + 2b = c T(d)=i 1a + 0b = a Note the cosets of the null space. Every element within a given coset has the same image. T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = b first column is T(a) = h = 1a + 2 b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = b first column is T(a) = h = 1a + 2 b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a second column is T(b) = e = 2a + 1 b T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = b first column is T(a) = h = 1a + 2 b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a second column is T(b) = e = 2a + 1 b T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e ( 2 ) 1 0a + 1b = b T(c)=h verify: e = 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
& 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} : 0a + 0b = i T(b)=e ( 2 ) 1 0a + 1b = b T(c)=h verify: e = 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) ( 2 ) 1 ( 1 ) 2 2a + 0b = f T(g)=i = = h 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e ( 2 ) 0a + 1b = b T(c)=h verify: d = 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) ( 2 ) ( ) 2a + 0b = f T(g)=i = = i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e 0a + 1b = 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} . 0a + 2b = c T(d)=i 1a + 0b = a T(e)=h 1a + 1b = g T(f)=e 1a + 2b = h ( 1 2 ) 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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} : 0a + 0b = i T(b)=e ( 1 2 ) 0a + 1b = b T(c)=h 0a + 2b = c T(d)=i 1a + 0b = 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 1a + 1b = g T(f)=e 1a + 2b = h 2a + 0b = f T(g)=i 2a + 1b = e T(h)=e 2a + 2b = d T(i)=i
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.