Linear Algebra I

Question 1

Define premultiplication of a matrix

Answer 1

If we premultiply B by A, we get AB.

Question 2

Define postmultiplication of a matrix

Answer 2

If we postmultiply B by A, we get BA.

Question 3

Define an inverse matrix

Answer 3

If B is the inverse of A, AB = BA = I

Question 4

Define a singular matrix

Answer 4

A matrix with no inverse.

Question 5

Define a matrix’s transpose

Answer 5

The transpose of a matrix is given by (A^T)ij = Aji

Question 6

Define a symmetric matrix

Answer 6

A square matrix where A^T = A

Question 7

Define a skew symmetric matrix

Answer 7

A square matrix where A^T = -A

Question 8

Define an orthogonal matrix

Answer 8

A matrix, A is orthogonal if A^(-1) = A^T

Question 9

Define a real vector space

Answer 9

V is a real vector space if V is non empty and there exists operations:
Addition: u + v ∈ V => u + v ∈ V
Scalar multiplication: For all α ∈ R and u ∈ V, αu ∈ V
satisfying the following axioms:
Addition is commutative, associative and there exists an additive identity and inverses.
Distributivity of scalar multiplication over vector addition
Distributivity of scalar multiplication over field addition.
Scalar multiplication interacts well with field multiplication.
There exists a scalar multiplicative identity.

Question 10

Define a vector subspace

Answer 10

W is a subspace of V if W is a subset of V that is closed under addition and scalar multiplication.

Question 11

Define a span of a set

Answer 11

Given a subset S {s_1, s_2, …, s_n} of a vector space V over a field F, <S> is the set of all linear combinations of S or the smallest subspace of V that contains all elements in S.</S>

Question 12

Define a spanning set for V

Answer 12

A spanning set for V is any set S for which <S> = V</S>

Question 13

Define linear independence

Answer 13

v_1, … ,v_m ∈ V are linearly independent if the only solution to (α_1)(v_1) + … + (α_m)(v_m) = 0, where α_i ∈ F is α_1 = … = α_m = 0.

Question 14

Define a basis

Answer 14

A linearly independent spanning set

Question 15

Give the Steinitz exchange lemma

Answer 15

Let V be a vector space over a field F. Take X = {v1, v2, . . . , vn}, a subset of V.
Suppose that u ∈ <X> but that u ∉ <X\{vi}> for some i.
Let Y = (X\{vi}) ∪ {u}
Then <Y> = <X></X></Y></X>

Question 16

Define the dimension of a vector space

Answer 16

The dimension of V is the size of any basis of V

Question 17

Define row rank

Answer 17

The row rank of A is the number of non zero rows in RRE(A)

Question 18

Define row space

Answer 18

The span of a matrix’s rows

Question 19

Define an internal direct sum

Answer 19

V = U ⊕ W if U, W ≤ V and UnW = {0v}

Question 20

Define a linear transformation

Answer 20

A map T: V -> W such that T(v1+v2) = T(v1) + T(v2) and T(αv) = αT(v) for all v in V and all α in F

Question 21

Define in invertible transformation

Answer 21

A linear transformation T: V -> W where there exists S: W -> V such that ST = id_V and TS = id_W

Question 22

Define the trace of a matrix

Answer 22

Trace(A) = a_(jj), summed for all j

Question 23

Define an isomorphism

Answer 23

An invertible linear map

Question 24

Define the kernal of a linear map

Answer 24

Ker(T) = {v ∈ V : T(v) = 0}

Question 25

Define the image of a linear map

Answer 25

Im(T) = {w ∈ W : ∃v ∈ V with T(v) = w}

Question 26

Define the nullity of T

Answer 26

dim(Ker(T))

Question 27

Define the rank of T

Answer 27

dim(Im(T))

Question 28

Define similar matrices

Answer 28

A and B are similar if there exists an invertible X such that A = X^-1BX

Question 29

Define a bilinear form

Answer 29

For V, a vector space over F, B:VxV -> F is a bilinear form if B(au + bv, w) = a(u,w) + b(v,w) and B(u, av + bw) = a(u,v) + b(u,w) for u, v, w in V and a, b in F

Question 30

Define a Gram matrix

Answer 30

Given v1, …, vk in V, the kxk matrix G defined by g_ij = B(vi,vj) is the Gram matrix for B with respect to v1, …, vk

Question 31

Define a symmetric bilinear form

Answer 31

B: VxV -> F is a symmetric bilinear form if B(v,w) = B(w,v) for all v, w in V

Question 32

Define positive definite

Answer 32

A matrix where B(v,v) ≥ 0 for all v in V and B(v,v) = 0 if and only if v = 0v

Question 33

Define an inner product

Answer 33

A symmetric bilinear form that is also positive definite

Question 34

Define a norm

Answer 34

For V, an inner product space with inner product <v1,v2>, the norm of V is the positive square root of <v,v>.

Question 35

Define a unitary matrix

Answer 35

A complex matrix A is unitary if AA* = AA = I, where A is the conjugate of A