Maths 2B

Question 1

What is an elementary matrix?

Answer 1

“An elementary matrix is any matrix that can be obtianed by performing an elementary row operation on an identity matrix.”

Question 2

What is a linear combination?

Answer 2

“A vector v is a linear combination of vectors v₁, v₂, …, v_k if there are scalars c₁, c₂, …, c_k such that v = c₁v₁ + c₂v₂ + … + c_kv_k.

The scalars are called coefficients of the linear combination.”

Question 3

What is a span/spanning set?

Answer 3

“If S = {v₁, v₂, …, v_k} is a set of vectors in ℝⁿ, then the set of all linear combinations of v₁, v₂, …, v_k is called the span of v₁, v₂, …, v_k and is denoted by span(v₁, v₂, …, v_k) or span(S).

If span(s) = ℝⁿ, then S is called a spanning set for ℝⁿ.”

Question 4

What is linear dependence and independence?

Answer 4

“A set of vectors v₁, v₂, …, v_k is linearly dependent if there are scalars c₁, c₂, … , c_k, at least one of which is not zero, such that v = c₁v₁ + c₂v₂ + … + c_kv_k = 0

A set of vectors that is not linearly dependent is called linearly independent.”

Question 5

What is a basis?

Answer 5

“A basis for a subspace S of ℝⁿ is a set of vectors S that

1. Spans S
2. Is linearly independent”

Question 6

What is a dimension?

Answer 6

“If S is a subspace of ℝⁿ, then the number of vectors in a basis for S is called the dimension of S, denoted dim S.”

Question 7

What is a null space?

Answer 7

“Let A be a m x n matrix. The null space of A is the subspace of ℝⁿ consisting of solutions of the homogeneous linear system Ax = 0. It is denoted by null(A).”

Question 8

What is a rowspace?

Answer 8

“The row space of an m x n matrix is the subspace row(A) of ℝⁿ spanned by the rows of A.”

Question 9

What is a column space?

Answer 9

“The column space of a m x n matrix A is the subspace col(A) of ℝ^m spanned by the columns of A.”

Question 10

What is the rank of a matrix?

Answer 10

“The rank of a matrix is the number of nonzero rows in its row echelon form.”

Question 11

What is the nullity?

Answer 11

“The nullity of a matrix A is the dimension of its null space and is denoted by nullity(A)”

Question 12

What is a transformation?

Answer 12

“A transformation T: ℝⁿ → ℝ^m is called a linear transformation

Question 13

What is the kernel of a linear transformation?

Answer 13

“Let T: V → W be a linear transformation. The kernel of T, denoted ker(T), is the set of all vectors in V that are mapped by T to 0 in W. That is,

ket(T) = { v in V: T(v) = 0}

Question 14

What is the range?

Answer 14

“The range of T, denoted range(T), T: V → W, is the set of all vectors in W that are images of vectors in V under T. That is,

range(T) = {T(v) : v in V}

= {w in W: w = T(v) for some v in V}”

Question 15

What is an isomorphism?

Answer 15

“A linear transformation T: V → W is called an isomorphism if it is one-to-one and onto. If V and W are two vector spaces such that there is an isomorphism from V to W, then we say that V is isomorphic to W and write V = W.”

Question 16

What is a smilar matrix?

Answer 16

” Let A and B be n x n matrices. We say that A is similar to B if there is an invertible n x n matrix P such that P^-1 AP = B. If A is similar to B we write A _~ B.”

Question 17

What is a diagonalisable matrix?

Answer 17

“An n x n matrix A is diagonalizable if there is a diagonal matrix D such that A is similar to D (A _~ D) - that is there is an invertible n x n matrix P such that P^-1AP = D.

Question 18

What is a diagonal matrix?

Answer 18

“A square matrix whose nondiagonal entries are all zero is called a diagonal matrix”

Question 19

What is an eigenvalue and subsequently an eigenvector?

Answer 19

“Let A be a n x n matrix. A scalar λ is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx.

Such a vector x is called an eigen-vector of A corresponding to λ.”

Question 20

What is an eigenspace?

Answer 20

“Let A be an n x n matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to λ, together with the zero vector, is called the eigenspace of λ and is denoted E_λ.”

Question 21

What is geometric multiplicity?

Answer 21

“Let us define the geometric multiplicity of an eigenvalue λ to be dim E_k, the dimension of its corresponding eigenspace.”

The geometric multiplicity is the number of linearly independent eigenvectors you can find for an eigenvalue.

Question 22

What is algebraic multiplicity?

Answer 22

‘The algebraic multiplicity of an eigenvalue is the multiplicity as a root of the characteristic polynomial.”

The algebraic multiplicity of an eigenvalue λ is the power m of the term (x − λ)^m in the characteristic polynomial.