MATH 817 Notes
JD Nir
[email protected]
www.math.unl.edu/∼jnir2/817.html
November 16, 2015
Let B be a basis of an F vector space V , and let W be any F -vector space. Then the function
HomF (V, W ) → Functions(B, W )
given by ϕ 7→ ϕ|B is bijective.
Say B = (v1 , v2 , . . . , vn ) is a basis of V and C = (w1 , w2 , . . . , wm ) is a basis of W .
1-1
1-1
HomF (V, W ) ←→ Functions(B, W ) ←→
onto
onto
ϕ
ϕ|B
7→
(g : B → W )
7→

a11


a
 21

 ..
 .

am1
Matm×n (F )

a12 · · · a1n
..
.. 

.
. 


..


.

am2 · · · amn
Notation
• Given V, W and B, C as above, for ϕ ∈ HomF (V, W ) MBC (ϕ) := [aij ]m×n , ϕ(vi ) = a1i w1 +
a2i w2 + · · · + ami wm .
 
λ1
 λ2 
 
• For v ∈ V , [v]B =  .  where v = λ1 b1 + · · · + λn bn
 .. 
λn
Prop Let V, W, B, C be as above.
1 V ∼
= F n via v 7→ [v]B
↑
isomorphism of vector space


λ1
P
 
λi vi ←[  ... 
λn
2 There is a bijection HomF (V, W ) ∼
= Matm×n (F ) given by ϕ 7→ MBC (ϕ)
3 ∀v ∈ V, ∀ϕ ∈ HomF (V, W )
[ϕ(v)]C = MBC (ϕ) · [v]B
4 If A = (u1 , u1 , . . . , up ) is a basis of another F -vector space U , then ∀ϕ ∈ HomF (V, W ), ∀ψ :
HomF (U, V ),
C
B
MatC
A (ϕ ◦ ϕ) = MatB (ϕ) · MatA (ψ)
↑
matrix multiplication
Note A, B, C are ordered bases in this proposition.
Cor 1: Matrix multiplication is associative.
1
JD Nir
[email protected]
MATH 817
Cor 2: HomF (V, V ) ∼
= Matn×n (F ) and also AutF (V ) ∼
= GLn (F )
↑
isomorphism of groups.
0
Similarity: Let B, B 0 are two ordered bases of V (#B = #B 0 < ∞). Let P = MBB (idv ), idV : V →
V is identity map.
Then
0
Prop
• P is invertible and P −1 = MBB0 (idV ). Why? MBB0 (idv ) · MBB (idv ) = MBB (idv ) = In and
0
MBB (idv ) · MBB0 (idv ) = In
• For any ϕ ∈ HomF (V, V ),
0
P · MBB (ϕ) · P −1 = MBB (idV ) · MBB (ϕ) · MBB0 (idV )
0
= MBB (idV · ϕ) · MBB0 (idV )
0
= MBB0 (idV ◦ ϕ ◦ idV )
0
= MBB0 (ϕ)
So, similarity of matricies amounts to a change of basis:
Given base B, B 0 of V , if ϕ ∈ HomF (U, V ), then
0
MBB (ϕ) ∼ MBB0 (ϕ)
↑
similarity
Equivalence Let V, W be two vector spaces, say B, B 0 are two ordered bases of V and C, C 0 are two
ordered bases of W .
0
∀ϕ ∈ Q · HomF (V, W ), MBC (ϕ) · P = MBC0 (ϕ)
where
P = MBB0 (idV )
.
0
Q = MCC (idw )
If A, A0 are two m × n matricies, then A and A0 are equivalent if
A0
m×n
=Q A P
m×n
for some Q ∈ GLn (F ), P ∈ GLn (F ).
Note: ∀Q ∈ GLn (F ), QA is obatained from A via element row operations.
∀P ∈ GLn (F ), AP is obtained from A via elementary column operations.
Def Say W is a subspace of a F -vector space V then V /W =
λv + W .
2
(V,+)
(W,+)
is a vector space via λ·(v +W ) :=