Chapter 1 Flashcards
Examples of fields?
- real numbers
- complex numbers
- integers modulo n
What is a field?
A set with the operators addition and multiplication
What 5 axioms do operators of a field have to satisfy?
F1) Commutativity
F2) Associativity
F3) Distributivity
F4) Identities
F5) Inverses
What is the field of 2 numbers called?
Modulo 2
What would a vector in (F₂)³ look like?
(1̅, 0̅, 1̅)
What is a vector space?
A vector space V over a field F is a set with 2 operations: addition and multiplication by scalars
What 5 axioms do operators of a vector space have to satisfy?
V1) Closure
V2) Commutativity and associativity of addition
V3) Distributivity and compatibility of scalar multiplication
V4) 1v = v
V5) There is a unique element, call it 𝟎 ⋲ V, such that 0𝐯 = 𝟎
What is a subspace?
A subspace of V is a subset U which contains 𝟎 and is closed under addition and scalar multiplication
What do you need to do to show something is not a subspace?
Find a counter-example
What is a linear span?
A linear span of a finite set of vectors S = {v₁, v₂, …, vn} in a vector space V is the set of all linear combinations of the vectors. That is:
span(S) = {ɑ₁v₁ + … + ɑnvn : ɑ₁ … ⋲ F}
How would you prove span(S) is a subspace of V? where S = {v₁, v₂, …, vn} where vi ⋲ V∀i
Check that:
- Contains zero element
- Addition holds
- Scalar multiplication holds
What are the 3 different types of subspace of R²?
- zero vector
- lines through the origin ( = R¹)
- planes through the origin ( = R²)
What is a spanning set?
S = {v₁, v₂, …, vn} is a spanning set of V if span(S) = V
we say that S spans V
What is span(2u) equal to?
span(u)
What is span(u₁, u₁ + u₂) equal to?
span(u₁, u₂)
What is span(u₁, u₂, u₂) equal to?
span(u₁, u₂)
What is linear dependence?
A set of vectors {𝐯₁, 𝐯₂, …, 𝐯n} are linearly dependent if
ɑ₁𝐯₁ + ɑ₂𝐯₂ + … + ɑn𝐯n = 𝟎 for some non zero a_is
What is a basis?
Let V be a vector space.
A finite set of vectors S = {𝐮₁, 𝐮₂, …, 𝐮n} is a basis of V if it spans V and is linearly independent
What is equivalent to the statement ‘S = {𝐮₁, 𝐮₂, …, 𝐮n} is a basis of V’?
Every 𝐮 ⋲ V can be uniquely written as 𝐮 = ɑ₁𝐮₁ + ɑ₂𝐮₂ + … + ɑn𝐮n
What is a bijection?
A 1-1 map
What is the definition of dimension?
If V has a basis with n elements, then V has dimension n, dim(V) = n
What is the Steinitz exchange lemma?
Let {𝐯₁, …, 𝐯n} span V, and {𝐮₁, …, 𝐮m} be a linearly independent subset of V. Then m < n and the set {𝐮₁, …, 𝐮m, 𝐯(m₊₁), …, 𝐯n} span V
(though you might have to reorder the 𝐯is)
What method to prove Steinitz exchange lemma?
Proof by induction
What is the relationship between the no. of elements in a LI set and the no. of elements in a spanning set?
No. of elements in LI set < spanning set
Do any 2 bases of V have the same number of elements?
Yes
If dimV = n, and you have n vectors in V, when will they form a basis for V?
If these span V or if they are linearly independent
When are the definitions for span, linear independence and bases equivalent?
For finite-dimensional vector spaces
W is a subset of V and dim(V) = n. What does this mean for dim(W)?
- dim(W) <= dim(V)
* if dim(W) = dim(V) then W=V
If matrix B can be obtained from matrix A by row operations, what implications does this have? (2 points)
- span of rows of A = spans of rows of B
2. rows of A are LD ⤄ rows of B are LD
Does every spanning set of V contain a basis for V?
Yes
Examples of commutativity? using a, b ⋲ F
- a + b = b + a
* a • b = b • a
Examples of associativity? using a, b, c ⋲ F
- a + (b + c) = (a + b) + c
* a • (b • c) = (a • b) • c
Example of distributivity? using a, b, c ⋲ F
a • (b + c) = a • b + a • c