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Vector Spaces and Linear Algebra
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Vector Spaces and Linear Algebra
Vector Spaces and Linear Algebra
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1
Question
What is the definition of a vector space over a commutative field K?
Page 4
Answer
(E, +, ·) where (E, +) is an abelian group and · : K × E → E satisfies compatibility properties.
2
Question
What are the group properties of the addition operation in a vector space?
Page 4
Answer
Associativity, commutativity, neutral element 0_E, and every element has an inverse.
3
Question
List the first four properties of scalar multiplication in a vector space.
Page 5
Answer
1. 1_K · x = x. 2. (λ + β) · x = λx + βx. 3. λ · (x + y) = λx + λy. 4. (λβ) · x = λ · (βx).
4
Question
What are elements of the field K called in vector spaces, and elements of E?
Page 5
Answer
Scalars for K, vectors for E.
5
Question
Give three examples of vector spaces.
Page 5
Answer
R as R-vector space, C as C-vector space, K[X] polynomials as K-vector space.
6
Question
State Proposition 1 for vector spaces.
Page 5
Answer
0·v = 0_E, (-1_K)·v = -v, α·v=0_E iff α=0 or v=0_E, etc.
7
Question
What is a linear combination of vectors u1,...,un over K?
Page 6
Answer
\(\\sum_{i=1}^n \\lambda_i u_i\) where \\lambda_i \\in K.
8
Question
Define a subspace F of vector space E over K using closure properties.
Page 6
Answer
F ≠ ∅, closed under addition x+y ∈ F, closed under scalar λx ∈ F.
9
Question
Are the three definitions of subspace equivalent?
Page 6
Answer
Yes, closed under +, scalar; linear combs; etc.
10
Question
Why does every subspace contain the zero vector 0_E?
Page 6
Answer
Take x ∈ F, -x ∈ F by closure, x + (-x) = 0_E ∈ F.
11
Question
Is the union of two subspaces a subspace?
Page 7
Answer
No, generally not (counterexample on page 7).
12
Question
Is F = {(x,y) ∈ R^2 | x + y = 0} a subspace of R^2?
Page 7
Answer
Yes, contains 0, closed under + and scalar mult.
13
Question
Why is F = {(x,y) ∈ R^2 | x=1} not a subspace?
Page 7
Answer
(0,0) ∉ F since 0 ≠ 1.
14
Question
Define linear independence of {u1,...,un}.
Page 7
Answer
\(\\sum \\lambda_i u_i = 0\\) implies all \\lambda_i = 0.
15
Question
Is {(1,0), (0,1)} linearly independent in R^2?
Page 7
Answer
Yes, α(1,0) + β(0,1) = (0,0) implies α=β=0.
16
Question
Why is {(2,2),(4,4)} linearly dependent?
Page 8
Answer
(4,4) = 2(2,2), non-trivial combo to zero.
17
Question
Properties of linearly independent sets from Prop 4.
Page 8
Answer
∅ lin ind, all distinct, cannot contain 0 unless singleton.
18
Question
What is the span of a set V in vector space F?
Page 8
Answer
All linear combinations of vectors in V.
19
Question
Give examples of spans in low dimensions.
Page 8
Answer
span{(1,i)}=C, span{(1,1)}=line x=y in R^2, span std basis = R^n.
20
Question
What is a basis for subspace F?
Page 9
Answer
Lin ind set that spans F.
21
Question
What are coordinates of u = ∑ λ_i v_i in basis {v1..vn}?
Page 9
Answer
The scalars λ1, ..., λn.
22
Question
Define dimension of vector space E.
Page 9
Answer
Number of vectors in any basis (Thm 2 all same). dim{0}=0.
23
Question
When is B={v1..vn} basis for R^n?
Page 9
Answer
Lin ind or spans R^n (Prop 5).
24
Question
State the incomplete basis theorem.
Page 9
Answer
Lin ind set of size r < n in dim n space extends to basis by adding n-r vectors.
25
Question
Define rank of vector set {v1..vp}.
Page 10
Answer
dim span{v1..vp}.
26
Question
From Prop 6, when rank{v1..vp}=p?
Page 10
Answer
Set linearly independent.
27
Question
In Ex11, rank{v1,v2,v3} where v1-v2+v3=0?
Page 10
Answer
2, since dependent, rank{v1,v2}=2.
28
Question
Define sum of subspaces F1 + F2.
Page 11
Answer
{x1 + x2 | x1∈F1, x2∈F2}.
29
Question
When is F1 + F2 a direct sum F1 ⊕ F2?
Page 11
Answer
F1 ∩ F2 = {0_E}.
30
Question
Prove unique decomposition in direct sum.
Page 11
Answer
x = x1+x2 = y1+y2 ⇒ x1-y1 = y2-x2 ∈ F1∩F2={0}.