MA106 - Linear Algebra

Question 1

Define what is meant by a Matrix

Answer 1

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.

Question 2

Define Addition of Matrices

Answer 2

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.

Question 3

Define Scalar Multiplication of Matrices

Answer 3

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.

Question 4

Define Multiplication of Matrices

Answer 4

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.

Question 5

Define what is meant by the Zero Matrix

Answer 5

The m × n zero matrix 0mn has all of its entries equal to 0.

Question 6

Define the Identity Matrix

Answer 6

The n × n identity matrix In = (aij ) has aii = 1 for 1 ≤ i ≤ n, but aij =0wheni̸=j.

Question 7

What are the Properties for a Matrix to be in Upper Echelon Form?

Answer 7

(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).

Question 8

What are the Properties for a Matrix to be in Row Reduced Form?

Answer 8

(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.

Question 9

Define the types of Elementary Row Operations.

Answer 9

(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

Question 10

Define what is meant by an Augmented Matrix.

Answer 10

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)

Question 11

Define the types of Elementary Column Operations

Answer 11

(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.

Question 12

What does a Matrix look like in Row and Column Reduced Form?

Answer 12

It is the Identity Is, surrounded by 0s from m-s to s or n-s to s

Question 13

What is another term for Row and Column Reduced Form?

Answer 13

Smith Normal Form

Question 14

Define what is meant by a Field.

Answer 14

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).

Question 15

What are the Axioms for Addition?

Answer 15

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.

Question 16

What are the Axioms for Multiplication?

Answer 16

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.

Question 17

What is the Axiom relating Addition & Multiplication?

Answer 17

D. (α+β).γ=α.γ+β.γ Ɐ α,β,γ∈S.

Question 18

What are the Axioms for Addition, Multiplication and Division?

Answer 18

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.

Question 19

Define what is meant by a Vector Space

Answer 19

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.

Question 20

Define what is meant by a Linear combination

Answer 20

A linear combination of v1, v2, . . . , vn is a vector of the form:
α1v1 + α2v2 + · · · + αnvn
for α1, α2, . . . , αn ∈ K.

Question 21

What is meant by the Trivial Linear Combination?

Answer 21

0v1 + · · · + 0vn = 0

Question 22

Define Linear Independence

Answer 22

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.

Question 23

Define what is meant by Spanning Vectors

Answer 23

A subset S ⊂ V spans V if every v ∈ V is a finite linear combination:
α1v1 + α2v2 +···+ αnvn
with v1,…,vn ∈S.

Question 24

Define what is meant by a Basis of a Vector Space

Answer 24

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}.

Question 25

Define the Coordinates of V wrt the basis: | v1, v2, ..., vn

Answer 25

If v = α1v1 + · · · + αnvn | The scalars α1, . . . , αn are the coordinates of v wrt the basis v1, . . . , vn.

Question 26

Define the Dimension of a Vector Space.

Answer 26

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.

Question 27

Define what is meant by a Subspace.

Answer 27

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 .

Question 28

Define W1 + W2 where W1, W2 are subspaces of V

Answer 28

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}.

Question 29

Define what is meant by a Linear Transformation.

Answer 29

``` 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. ```

Question 30

Define the Identity Map.

Answer 30

The Identity Map IV : V → V by: | IV (v) = v Ɐ v∈V.

Question 31

Define the Zero Map.

Answer 31

The Zero Map 0U,V : U → V by: | 0U,V(u)=0V Ɐ u∈U.

Question 32

Define Addition of Linear Maps

Answer 32

T1 + T2 : U → V by the rule | (T1 + T2)(u) = T1(u) + T2(u) Ɐ u ∈ U.

Question 33

Define Scalar Multiplication of Linear Maps

Answer 33

αT1 : U → V by the rule | (αT1)(u) = αT1(u) Ɐ u ∈ U.

Question 34

Define Composition of Linear Maps

Answer 34

T2T1 : U → W by | (T2T1)(u) = T2(T1(u)) Ɐ u ∈ U.

Question 35

Define what is meant by the Image of a Linear Map

Answer 35

The image of T, denoted im(T) is the set of v∈V s.t. v=T(u) for some u∈U.

Question 36

Define What is Meant by the Kernel of a Linear Map

Answer 36

The kernel of T , denoted ker(T) is the set of u ∈ U s.t. T(u) = 0V .

Question 37

Define Rank and Nullity of a Linear Map

Answer 37

(i) dim(im(T)) is called the rank of T; | (ii) dim(ker(T)) is called the nullity of T.

Question 38

Define Row Space and Row Rank of a matrix A.

Answer 38

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.

Question 39

Define what it means for a Linear Map/Matrix to be Invertible

Answer 39

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

Question 40

Define the Determinant of a Matrix

Answer 40

det(A) = 􏰌Sum(over φ∈Sn): | sign(φ)a1φ(1).a2φ(2) . . . anφ(n).

Question 41

Define what is meant for a Matrix to be Upper Triangular

Answer 41

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.

Question 42

Define what is meant for a Matrix to be Diagonal

Answer 42

A matrix is diagonal if all entries not on the main diagonal are zero; aij =0 Ɐ i̸=j.

Question 43

Define the Transpose of a Matrix

Answer 43

The Transpose A^T of A is the n×m matrix (bij): | where bij =aji for1≤i≤n, 1≤j≤m.

Question 44

Define the 3 Elementary Matrices

Answer 44

``` 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) ```

Question 45

Define a Minor of a Matrix

Answer 45

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.

Question 46

Define a Cofactor of a Matrix

Answer 46

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.

Question 47

Define what is meant by the Adjugate Matrix

Answer 47

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.

Question 48

Define Similarity of Matrices

Answer 48

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.

Question 49

Define what is meant by Diagonisable

Answer 49

A matrix which is similar to a diagonal matrix is said to be diagonalisable.

Question 50

Give the definition of Eigenvectors/Eigenvalues of a Linear Map

Answer 50

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.

Question 51

Give the definition of Eigenvectors/Eigenvalues of a Matrix

Answer 51

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.

Question 52

What is The Characteristic Equation/Polynomial for a Matrix

Answer 52

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.