MA106 - Linear Algebra Flashcards
Define what is meant by a Matrix
An m × n matrix A over R is an m × n rectangular array of real numbers. The entry in row i and column j is often written aij. We write A = (aij) to make things clear.
Define Addition of Matrices
Let A = (aij ) and B = (bij ) be two m × n matrices over R. We define A+B to be the m×n matrix C = (cij ), where cij = aij +bij for all i, j.
Define Scalar Multiplication of Matrices
Let A = (aij) be an m × n matrix over R and let β ∈ R. We define the scalar multiple βA to be the m×n matrix C = (cij), where cij = βaij for all i,j.
Define Multiplication of Matrices
Let A = (aij ) be an l × m matrix over R and let B = (bij) be an m × n matrix over K. The product AB is an l × n matrix C = (cij) where, for 1 ≤ i ≤ l and 1 ≤ j ≤ n, cij =sum(k=1 to m) aikbkj =ai1b1j +ai2b2j +···+aimbmj.
Define what is meant by the Zero Matrix
The m × n zero matrix 0mn has all of its entries equal to 0.
Define the Identity Matrix
The n × n identity matrix In = (aij ) has aii = 1 for 1 ≤ i ≤ n, but aij =0wheni̸=j.
What are the Properties for a Matrix to be in 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. In other words, ai,c(i) = 1 for all i ≤ s.
(iii) The first non-zero entry of each row is strictly to the right of the first non-zero
entry of the row above: that is, c(1) < c(2) < ··· < c(s).
What are the Properties for a Matrix to be in Row Reduced 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. In other words, ai,c(i) = 1 for all i ≤ s.
(iii) The first non-zero entry of each row is strictly to the right of the first non-zero
entry of the row above: that is, c(1) < c(2) < ··· < c(s).
(iv) If row i is non-zero, then all entries both above and below the first non-zero entry of row i are zero: ak,c(i) = 0 for all k ̸= i.
Define the types of Elementary Row Operations.
(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
Define what is meant by an Augmented Matrix.
For the system Ax = b of m equations in n unknowns.
If A is an m x n matrix, b=(bi1) The augmented matrix is the m x (n+1) matrix with
(ai1, ai2, …, ain | bi)
Define the types of Elementary Column Operations
(C1) For some i ̸= j, add a multiple of cj to ci.
(C2) Interchange two columns.
(C3) Multiply a column by a non-zero scalar.
What does a Matrix look like in Row and Column Reduced Form?
It is the Identity Is, surrounded by 0s from m-s to s or n-s to s
What is another term for Row and Column Reduced Form?
Smith Normal Form
Define what is meant by a Field.
A field is the data of a set S, two special elements 0 ̸= 1 ∈ S, and two maps S × S → S, called addition and multiplication, respectively satisfying certain axioms. We write α + β for the result of applying the addition map (α, β), and αβ for the result of applying the multiplication map to (α,β).
What are the Axioms for Addition?
A1. α+β=β+α Ɐ α,β∈S.
A2. (α+β)+γ=α+(β+γ) Ɐ α,β,γ∈S.
A3. ∃ 0∈S st. α+0=0+α=α Ɐ α∈S.
A4. Ɐ α ∈ S ∃ −α ∈ S s.t. α+(−α) = (−α)+α = 0.
What are the Axioms for Multiplication?
M1. α.β=β.α Ɐ α,β∈S.
M2. (α.β).γ = α.(β.γ) Ɐ α, β, γ ∈ S.
M3. ∃ 1∈S s.t. α.1=1.α=α Ɐ α∈S.
M4. Ɐ α ∈ S with α ̸= 0, ∃ α^−1 ∈ S s.t.
α.α^−1 = α^−1.α = 1.
What is the Axiom relating Addition & Multiplication?
D. (α+β).γ=α.γ+β.γ Ɐ α,β,γ∈S.
What are the Axioms for Addition, Multiplication and Division?
A1. α+β=β+α Ɐ α,β∈S.
A2. (α+β)+γ=α+(β+γ) Ɐ α,β,γ∈S.
A3. ∃ 0∈S st. α+0=0+α=α Ɐ α∈S.
A4. Ɐ α ∈ S ∃ −α ∈ S s.t. α+(−α) = (−α)+α = 0.
M1. α.β=β.α Ɐ α,β∈S.
M2. (α.β).γ = α.(β.γ) Ɐ α, β, γ ∈ S.
M3. ∃ 1∈S s.t. α.1=1.α=α Ɐ α∈S.
M4. Ɐ α ∈ S with α ̸= 0, ∃ α^−1 ∈ S s.t.
α.α^−1 = α^−1.α = 1.
D. (α+β).γ=α.γ+β.γ Ɐ α,β,γ∈S.
Define what is meant by a Vector Space
A vector space over a field K is a set V with 2 basic operations, Addition and Scalar Multiplication, satisfying certain requirements. Thus Ɐ u,v∈V, u+v∈V is defined ,and Ɐ α∈K, αv∈V is defined. For V to be called a vector space, the following axioms must be satisfied Ɐ α, β ∈ K and Ɐ 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.
Define what is meant by a Linear combination
A linear combination of v1, v2, . . . , vn is a vector of the form:
α1v1 + α2v2 + · · · + αnvn
for α1, α2, . . . , αn ∈ K.
What is meant by the Trivial Linear Combination?
0v1 + · · · + 0vn = 0
Define Linear Independence
Let V be a vector space over the field K. The vectors v1, . . . , vn ∈ V are linearly dependent if there is a non-trivial linear combination of v1, . . . , vn that equals the zero vector.
The vectors v1,…,vn are linearly independent if they are not linearly dependent.
A set S ⊂ V is linearly independent if every finite subset of S is linearly independent.
Define what is meant by Spanning Vectors
A subset S ⊂ V spans V if every v ∈ V is a finite linear combination:
α1v1 + α2v2 +···+ αnvn
with v1,…,vn ∈S.
Define what is meant by a Basis of a Vector Space
A subset S ⊂ V is a basis of V if S is linearly independent, and spans V . If the basis S is finite we also choose an ordering of its vectors, and write S = {v1,…,vn}.
Define the Coordinates of V wrt the basis:
v1, v2, …, vn
If v = α1v1 + · · · + αnvn
The scalars α1, . . . , αn are the coordinates of v wrt the basis v1, . . . , vn.
Define the Dimension of a Vector Space.
The number n of vectors in a basis of the finite-dimensional vector space V is called the dimension of V and we write dim(V ) = n.
Define what is meant by a Subspace.
A subspace of V is a non-empty subset W ⊆ V such that
(i) W isclosedunderaddition:
u,v∈W ⇒u+v∈W;
(ii) W is closed under scalar multiplication:
v ∈ W, α ∈ K ⇒ αv ∈ W .
Define W1 + W2 where W1, W2 are subspaces of V
W1 + W2 is the set of vectors v∈V s.t.
v=w1+w2 for some w1 ∈ W1, w2 ∈ W2.
W1+W2 ={w1 + w2 |w1 ∈ W1,w2 ∈ W2}.
Define what is meant by a Linear Transformation.
Let U,V be two vector spaces over the field K. A linear map T from U to V is a function T : U → V s.t.: (i) T(u1+u2)=T(u1)+T(u2) Ɐ u1,u2 ∈U; (ii) T(αu)=αT(u) Ɐ α∈K , u∈U.
Define the Identity Map.
The Identity Map IV : V → V by:
IV (v) = v Ɐ v∈V.
Define the Zero Map.
The Zero Map 0U,V : U → V by:
0U,V(u)=0V Ɐ u∈U.
Define Addition of Linear Maps
T1 + T2 : U → V by the rule
(T1 + T2)(u) = T1(u) + T2(u) Ɐ u ∈ U.
Define Scalar Multiplication of Linear Maps
αT1 : U → V by the rule
(αT1)(u) = αT1(u) Ɐ u ∈ U.
Define Composition of Linear Maps
T2T1 : U → W by
(T2T1)(u) = T2(T1(u)) Ɐ u ∈ U.
Define what is meant by the Image of a Linear Map
The image of T, denoted im(T)
is the set of v∈V s.t.
v=T(u) for some u∈U.
Define What is Meant by the Kernel of a Linear Map
The kernel of T , denoted ker(T)
is the set of u ∈ U s.t.
T(u) = 0V .
Define Rank and Nullity of a Linear Map
(i) dim(im(T)) is called the rank of T;
(ii) dim(ker(T)) is called the nullity of T.
Define Row Space and Row Rank of a matrix A.
The row space of A is the subspace of K^n spanned by the rows r1,…,rm of A.
The row rank of A is equal to the dimension of the row space of A.
Equivalently, the row rank of A is equal to the size of the largest linearly independent subset of r1, . . . , rm.
Define what it means for a Linear Map/Matrix to be Invertible
Let T : U → V be a linear map with corresponding m × n matrix A.
If ∃ T^−1:V →U s.t. TT^−1 =IV andT^−1T =IU
Then T is Invertible, T^-1 is the Inverse
A^−1 is the (n×m) matrix of T^−1, then:
AA^−1 = Im and A^−1A = In.
A is Invertible with Inverse A^-1
Define the Determinant of a Matrix
det(A) = Sum(over φ∈Sn):
sign(φ)a1φ(1).a2φ(2) . . . anφ(n).
Define what is meant for a Matrix to be Upper Triangular
A matrix is called upper triangular if all of its entries below the main diagonal are zero;
(aij) is upper triangular if aij = 0 Ɐ i > j.
Define what is meant for a Matrix to be Diagonal
A matrix is diagonal if all entries not on the main diagonal are zero;
aij =0 Ɐ i̸=j.
Define the Transpose of a Matrix
The Transpose A^T of A is the n×m matrix (bij):
where bij =aji for1≤i≤n, 1≤j≤m.
Define the 3 Elementary Matrices
E(n)1λ,i,j (i ̸= j) is the an n × n matrix equal to the identity, but with an additional non-zero entry λ in the (i,j) position. Corresponds to (R1)
E(n)2i,j is the n × n identity matrix with its ith and jth rows interchanged. Corresponds to (R2)
E(n)3λ,i (where λ ̸= 0) is the n × n identity matrix with its (i, i) entry replaced by λ. Corresponds to (R3)
Define a Minor of a Matrix
Let A = (aij) be an n×n matrix. Let Aij be the (n−1)×(n−1) matrix obtained from A by deleting the ith row and the jth column of A. Now let Mij = det(Aij). Then Mij is called the (i,j)th minor of A.
Define a Cofactor of a Matrix
We define cij to be equal to Mij if i + j is even, and to −Mij if i + j is odd. Or, more concisely,
cij = (−1)i+jMij = (−1)i+j det(Aij).
Then cij is called the (i,j)th cofactor of A.
Define what is meant by the Adjugate Matrix
Let A be an n × n matrix. We define the adjugate matrix adj(A) of A to be the n × n matrix of which the (i, j)th element is the cofactor cji. In other words, it is the transpose of the matrix of cofactors.
Define Similarity of Matrices
Two n × n matrices, A,B over K are said to be similar if there exists an n × n invertible matrix P with B = P−1AP.
Define what is meant by Diagonisable
A matrix which is similar to a diagonal matrix is said to be diagonalisable.
Give the definition of Eigenvectors/Eigenvalues of a Linear Map
Let T : V → V be a linear map, where V is a vector space over K. Suppose that for some non-zero vector v ∈ V and some scalar λ ∈ K, we have T (v) = λv. Then v is called an eigenvector of T , and λ is called the eigenvalue of T corresponding to v.
Give the definition of Eigenvectors/Eigenvalues of a Matrix
Let A be an n×n matrix over K. Suppose that, for some non-zero column vector v ∈ Kn,1 and some scalar λ ∈ K, we have Av = λv. Then v is called an eigenvector of A, and λ is called the eigenvalue of A corresponding to v.
What is The Characteristic Equation/Polynomial for a Matrix
For an n × n matrix A, the equation
det(A − xIn ) = 0 is called the characteristic equation of A, and
det(A − xIn) is called the characteristic polynomial of A.
