Definitions

Question 1

Matrix

Answer 1

For m,n>=1 an m x n matrix is a rectangular array with m rows and n columns

Question 2

Row/column vector

Answer 2

A row vector is a 1 x n matrix. A column vector is an m x 1 matrix

Question 3

Scalar (in terms of matrices)

Answer 3

The entries of a matrix are called a scalar. They come from a field, usually F

Question 4

Addition and scalar multiplication of matrices

Answer 4

Addition: let A=(aij) and B=(bij) be m x n matrices. We define A+B=(cij) to have (i,j) entry cij=aij+bij
We describe this addition as coordinatewise

Scalar multiplication: Let A=(aij) be an m x n matrix over F and p be in F. We define pA to be the m x n matrix with (i,j) entry paij

Question 5

Commute (in terms of matrices)

Answer 5

AB=BA

Question 6

Upper triangular matrix

Answer 6

If aij=0 whenever i>j

So the matrix only has elements in and above the diagonal

Question 7

Lower triangular matrix

Answer 7

If aij=0 whenever i

Question 8

Invertible (in terms of matrices)

Answer 8

If there exists B st.

AB=In=BA

Question 9

Orthogonal matrix

Answer 9

AB=In=BA

Where B is the transpose of A

Question 10

Unitary matrix

Answer 10

AB=In=BA

Where B is the transpose of the complex conjugate of A

Question 11

Augmented matrix A|b

Answer 11

The m x n matrix A with the matrix b a joined as the (n+1)th column

Question 12

Echelon form

Answer 12

1) Leading coeffs are 1
2) Leading entries of lower rows occur to the right of leading entries of higher rows
3) Zero rows, if any, appear below any non-zero rows

Question 13

Determined and free variables

Answer 13

Let E|d be an augmented matrix where E is in echelon form. A variable Xi is determined if there exists a leading coeff in column i

Otherwise Xi is free

Question 14

Reduced row echelon form

Answer 14

A matrix is in reduced row echelon form if it is in echelon form and if all columns containing the leading entry of a row has all other entries 0

Question 15

Elementary matrix

Answer 15

For an ERO on an m x n matrix we define the corresponding elementary matrix to be the result of applying that ERO to Im

Question 16

Vector space

Answer 16

A vector space of F is a non-empty set V together with a map
VxV->V given by (v,v’)|->v+v’ and a map
FxV->V given by (p,v)|->pv
That satisfy the vector space axioms

Question 17

Vector space axioms

Answer 17

1) u+v=v+u (addition is commutative)
2) u+(v+w)=(u+v)+w (addition is associative)
3) there is 0v such that v+0v = v = 0v+v (existence of additive identity)
4) there is w such that v+w=0v (existence of additive inverse)
5) p(u+v)=pu+pv (distributivity of scalar multiplication over vector addition)
6) (p+q)v=pv+qv (distributivity of scalar multiplication of field addition)
7) (pq)v=p(qv) (scalar multiplication interacts well with field multiplication)
8) 1v=v (identity for scalar multiplication)

Question 18

Subspace

Answer 18

A subspace of V is a non-empty subset of V that is closed under addition and scalar multiplication

Question 19

Proper subspace

Answer 19

A subspace of V other than V

Question 20

Linear combination

Answer 20

Take u1,u2,…,um in V

A linear combination of u1,…,um is a vector
a1u1+…+amum for some a1,…,am in F

Question 21

Span

Answer 21

We define the span of u1,…um to be {a1u1+…+amum:a1,…,am in F

Question 22

Spanning set

Answer 22

A set S is a spanning set for V if Span(S)=V

We say S spans V

Question 23

Linearly dependant

Answer 23

We say that v1,…,vm in V is linearly dependant if there exists a1,…am not all 0 in F such that
a1v1+…+amvm=0

Question 24

Linearly independent

Answer 24

If for v1,…,vm in V and
a1,…,am in F

v1,…,vm are linearly independent if the only way
a1v1+…+amvm=0 is if a1=…=am=0

Question 25

Basis

Answer 25

A basis of V is a linearly independent spanning set

Question 26

Dimension

Answer 26

For a finite dimensional vector space V. The dimension of V is the size of any basis of V

Question 27

Row space

Answer 27

For an mxn matrix A over F. We define the row space of A tk be the span of the subset of F^n consisting of the rows of A

Question 28

Direct sum

Answer 28

For subspaces U,W of V. We say that V is the direct sum of U and W if
U+W=V and
U∩W = {0v}

Question 29

Direct compliment

Answer 29

For subspaces U, W of V

We say that U is the direct compliment of W in V if V is the direct sum of U and W

Question 30

Column space

Answer 30

For an mxn matrix A over F. We define the column space of A tk be the span of the subset of F^n consisting of the columns of A

Question 31

Row Rank

Answer 31

dim(rowsp(A))

Question 32

Column Rank

Answer 32

dim(columnsp(A))

Question 33

Linear transformation / map

Answer 33

Let V,W be vector spaces over F. We say that a map T:V->W is linear if
I) T(v1+v2)=T(v1)+T(v2) for all v1,v2 in V
II) T(Pv)=PT(v) for all v in V and P in F

Question 34

Invertible (in terms of linear maps)

Answer 34

Let T:V->W
We say T is invertible if there is a linear transformation S:W->V such that
ST=idv and TS=idw (where idv and idw are the identity maps on V and W respectively)

T is a function so has a unique inverse so no ambiguity in writing T^-1
(See intro to uni maths course)

Question 35

Kernel (or null space)

Answer 35

ker(T)={v∈V:T(v)=0w}

All the things that are mapped to 0

Question 36

Image (of a linear map)

Answer 36

Im(T)={T(v):v∈V}

Question 37

Nullity

Answer 37

For a finite dimensional V. Let T:V->W

We define null(T)=dim(kerT)

Question 38

Rank (of a linear map)

Answer 38

For a finite dimensional V. Let T:V->W

We define rank(T)=dim(ImT)

Question 39

A matrix for a linear transformation T:V->W with respect to specific ordered bases for V, W.

Answer 39

Let v1,…,vn be a basis for V, Let w1,…,wm be a basis for W. The matrix for T with respect to these bases is the mxn matrix where the jth column corresponds to the following coefficients of the basis of W.
T(vj)=a1jw1+a2jw2+,,,+amjwm

Question 40

Similar Matrices

Answer 40

A and B (∈Mnxn(F)) are similar if there exists and invertible nxn P such that. B=P(^-1)AP

Question 41

Rank of a matrix

Answer 41

For an n x m matrix A the rank of A is the rowrank of A

Question 42

Bilinear Form

Answer 42

For a vector space V over F, a bilinear form is a function, often written [v,v’] (inequalities don’t work) : VxV->F such that:

(i) [a1v1+a2v2,v3]=a1[v1,v3]+a2[v2,v3] (linear in first variable), and
(ii) [v1,a2v2+a3v3]=a2[v1,v2]+a3[v1,v3] (linear in second variable)

Question 43

Gram Matrix

Answer 43

Take v1,…,vn in V. The Gram matrix of v1,…,vn with respect to is the nxn matrix ([vi,vj]). So the (i,j)th element is [vi,vj]

Question 44

Symmetric bilinear form

Answer 44

We say that a bilinear form [v,v’] :VxV->F is symmetric if

[v1,v2]=[v2,v1] for all v1,v2 in F

Question 45

Positive definite bilinear form

Answer 45

REAL VECTOR SPACE ONLY

We say that a bilinear form [v,v’] :VxV->F is positive definite if [v,v]>= 0 for all v in V. [v,v]=0 iff v=0

Question 46

Inner Product

Answer 46

An inner product on a real vector space V is a positive definite symmetric bilinear form on V

Question 47

Inner Product Space

Answer 47

We say that a real vector space is an inner product space if it is equipped with an inner product,

Question 48

Norm

Answer 48

Let V be a real inner product space. For v in V, we define the norm of v to be ||v||:= sqrt([v,v])

Question 49

Orthonormal Set

Answer 49

Let V be an inner product space. We say that {v1,…,vn} a subset of V is an orthonormal set if for all i, j we have
[vi,vj]=δij (1 if i=j, 0 if i/=j)

Question 50

Sesquilinear Form

Answer 50

Let V be a complex vector space. A function [v,v’] : VxV->C is a sesquilinear form if

(i) [a1v1+a2v2,v3]= a1[v1,v3] + a2[v2,v3]
(ii) [v1,v2]= complex conjugate of [v2,v1]

Question 51

Positive definite sesquilinear form

Answer 51

We say a sesquilinear form is positive definite if [v,v]>=0 for all v with =0 iff v=0

Question 52

Complex inner product space

Answer 52

A complex inner product space is a complex vector space equipped with a positive definite sesquilinear form