Linear Algebra // Flashcards
(50 cards)
Field
A set S on which addition and multiplication are defined is called a field if it satisfies each of the axioms A1-4, M1-4 and D, and if, in addition 1 ≠ 0
Vector space
A vector space over a field K is a set V which has two basic operations, addition and scalar multiplication. Thus for every pair u,v ∈ V, u + v ∈ V is defined, and for every α ∈ K, αv ∈ V is defined. The following axioms are satisfied for all α,β ∈ K and all u,v ∈ V
i) vector addition satisfies axioms A1-4
ii) α(u + v) = αu + αv
iii) (α + β)v = αv + βv
iv) (αβ)v = α(βv)
v) 1v = v
Scalar
An element of the field K
Linearly dependent
Let V be a vector space over the field K. The vectors v1, v2, … vn are said to be linearly dependent if there exist scalars α1, α2, …, αn ∈ K, not all 0, such that α1v1 + α2v2 + … + αnvn = 0
Linearly independent
v1, v2, … vn are linearly independent if the only scalars α1, α2, …, αn ∈ K that satisfy α1v1 + α2v2 + … + αnvn = 0 are α1 = α2 = … = αn = 0
Linear combination
Vectors of the form α1v1 + α2v2 + … + αnvn for α1, α2, …, αn ∈ K are called linear combinations of v1, v2, …, vn
Span
The vectors v1, v2, …, vn in V span V if every vector v ∈ V is a linear combination α1v1 + α2v2 + … + αnvn of v1, …, vn
Basis
The vectors v1, …, vn in V form a basis of V if they are linearly independent and span V
Subspace
A subspace of V is a non-empty subset W ⊂ V such that where u, v ∈ W and α ∈ K, u + v ∈ W and αv ∈ W
The set W1 + W2
Let W1, W2 be subspaces of the vector space V. Then W1 + W2 = {w1 + w2 | w1 ∈ W1, w2 ∈ W2}
Linear map
Let U, V be two vector spaces over the same field K. A linear map T from U to V is a function T: U → V such that
i) T(u1 + u2) = T(u1) + T(u2) for all u1,u2 ∈ U;
ii) T(αu) = αT(u) for all α ∈ K and u ∈ U
Image
Let T: U → V be a linear map. The image of T, written as im(T) is defined to be the set of vectors v ∈ V such that v = T(u) for some u ∈ U
Kernel
Let T: U → V be a linear map. The kern of T, written as ker(T), is defined to be the set of vectors u ∈ U such that T(u) = 0v
Rank
dim(im(T))
Nullity
dim(ker(T))
Non-singular linear map
A linear map T such that dim(U) = dim(V) = n, such that T is bijective, rank(T) = n and nul(T) = 0
Addition on linear maps T1 + T2: U → V
(T1 + T2)(u) = T1(u) + T2(u) for u ∈ U
Scalar multiplication αT1: U → V
(αT1)(u) = αT1(u) for u ∈ U
Composition T2T1: U → W, where T1: U → V and T2: V → W
(T2T2)(u) = T2(T1(u)) for u ∈ U
Hom_K(U, V)
Hom_K(U,V) = {T: U → V | T is linear}
Multiplication of matrices AB
Let A = (αij) be an lxm matrix over K and let B = (βij) be an mxn matrix over K. The product AB = C = (γij) is an lxn matrix where for 1 ≤ i ≤ l and 1 ≤ j ≤ n, γij = Σ(k=1,m) αik*βkj
note: #cols of A = #rows of B
Nullspace of a matrix A
Given a matrix A, the set of column vectors x ∈ K^n,1 solving Ax = 0 is the nullspace of A
Elementary row operations R1, R2, R3
R1 for some i ≠ j, add a multiple of rj to ri
R2 interchange two rows
R3 multiply a row by a non-zero scalar
Upper echelon form
i) all zero rows are below all non zero rows
ii) let r1,…,rs be the non-zero rows. Then each ri with 1 ≤ i ≤ s has 1 as its first non-zero entry
iii) c(1) < c(2) < … < c(s)
iv) α_k,c(i) = 0 for all k > i
