Linear Algebra I Flashcards
Define premultiplication of a matrix
If we premultiply B by A, we get AB.
Define postmultiplication of a matrix
If we postmultiply B by A, we get BA.
Define an inverse matrix
If B is the inverse of A, AB = BA = I
Define a singular matrix
A matrix with no inverse.
Define a matrix’s transpose
The transpose of a matrix is given by (A^T)ij = Aji
Define a symmetric matrix
A square matrix where A^T = A
Define a skew symmetric matrix
A square matrix where A^T = -A
Define an orthogonal matrix
A matrix, A is orthogonal if A^(-1) = A^T
Define a real vector space
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.
Define a vector subspace
W is a subspace of V if W is a subset of V that is closed under addition and scalar multiplication.
Define a span of a set
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>
Define a spanning set for V
A spanning set for V is any set S for which <S> = V</S>
Define linear independence
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.
Define a basis
A linearly independent spanning set
Give the Steinitz exchange lemma
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>
Define the dimension of a vector space
The dimension of V is the size of any basis of V
Define row rank
The row rank of A is the number of non zero rows in RRE(A)
Define row space
The span of a matrix’s rows
Define an internal direct sum
V = U ⊕ W if U, W ≤ V and UnW = {0v}
Define a linear transformation
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
Define in invertible transformation
A linear transformation T: V -> W where there exists S: W -> V such that ST = id_V and TS = id_W
Define the trace of a matrix
Trace(A) = a_(jj), summed for all j
Define an isomorphism
An invertible linear map
Define the kernal of a linear map
Ker(T) = {v ∈ V : T(v) = 0}
Define the image of a linear map
Im(T) = {w ∈ W : ∃v ∈ V with T(v) = w}
Define the nullity of T
dim(Ker(T))
Define the rank of T
dim(Im(T))
Define similar matrices
A and B are similar if there exists an invertible X such that A = X^-1BX
Define a bilinear form
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
Define a Gram matrix
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
Define a symmetric bilinear form
B: VxV -> F is a symmetric bilinear form if B(v,w) = B(w,v) for all v, w in V
Define positive definite
A matrix where B(v,v) ≥ 0 for all v in V and B(v,v) = 0 if and only if v = 0v
Define an inner product
A symmetric bilinear form that is also positive definite
Define a norm
For V, an inner product space with inner product <v1,v2>, the norm of V is the positive square root of <v,v>.
Define a unitary matrix
A complex matrix A is unitary if AA* = AA = I, where A is the conjugate of A
