Advanced Linear Algebra Final

Final

Span of vectors v1 ,v2, . . . ,v n (also the term spanning/generating/complete)

Let V be a vector space over F. Let v1, …, vk ∈ V. The __ of v1…vk is the set of all linear combinations of v1…vk.

basis

A ___ for a vector space V over F is a set B={v1…vk} a subset of V such that for all v ∈ V there exsists a unique scalar alpha1…alphak ∈ F such that v=v1 alpha1+…+vk alphak

linearly independent, linearly dependent

Vectors {v1…vk} are a subset of V are ___ if: The only linear combination of v1…vk yielding the 0 vector is the “trivial” linear combination of 0v1+…+0vk.

v1…vk ∈ V are linearly dependent if they are not ___ or if there exists a non-trivial combination of v1…vk giving the zero vector.

subspace

Let V be a vector space over F. A ___ is a subset W ⊆ V that is also a vector space under the same addition and scalar multiplication in V.

Linear transformation

Let V, W be vector spaces (over the same field F). A transformation T : V →W is called ___ if

1. T(u+v) = T(u)+T(v) ∀u,v ∈ V;

2. T(αv) = αT(v) for all v ∈ V and for all scalars α ∈ F.

image

The ___ of T is Im(T) := { w ∈ W | ∃ v ∈ V such that T(v)=w} = {T(v) | v ∈ V} (Range of T)

kernel

The ___ of a linear map T: V —> W between two vector spaces V and W is the set of all vectors in V that are mapped to the zero vector in W.

invertible matrix, inverse of matrix

A matrix A ∈ M_{m x n}(ℝ) is ___ if ∃ B ∈ M_{m x n}(ℝ) such that AB = I_m and BA = I_n.

isomorphism

T: V —> W is an ___ if it is both injective and surjective and a linear transformation.

rank

The ___ of A is the dimension of the Colum Space. Rank A=dim(col(A))=dim(Im T_A)

Coordinate vector with respect to a basis

The map T_B is called a coordinate map with respect to B for v ∈ V, T_B(v) is called a coordinate vector for v with respect to B =: [v]_B

Eigenvalue

T: V—> V is a linear transformation. Then an eigenvector is a nonzero v in V s.t there exists lambda in F with T(v)= lambda v. Then lambda is an ___ for T(wrt v).

Eigenvector

A _____ of A is a nonzero vector v in F^n such that there exists a scalar satisfying Av=lambda v.

Eigenspace

The _____ is comprised of zero vectors and all eigenvectors for lambda.

Characteristic polynomial, characteristic equation

The _____ for A is det(A - lambda In). The _____ for A is det(A - lambda In)=0

sum of vector spaces

V is a v.s over F with w1, w2 subspaces of V. Then w1+w2 := {x+y | x in w1 and y in w2} is another subspace of V called the ___.

direct sum of vector spaces

V is called a ___ of w1, w2 if V = w1+w2 and for every x in V there exists a unique w1 in w1 and w2 in w2 s.t x = w1+w2.

diagonalizable matrix

A in M_nxn (F) is a ___ is there exsists a diagonal matrix D and an invertiable matrix P s.t D=P^-1AP (So A=PDP^-1)

Inner Product

Let V be a v.s over R. An ___ is a function <,>: VxV —> R (v,w)—> <v,w> s.t for every v,w,u in V and alpha, beta in R

1. Positive Definite: <v,v> >= 0 with equality iff v=0

2. Symmetric: <x,y> = <y,x>

3. Bi linearity: <alpha v+ beta w, u> = alpha<v,u> + beta<w,u>

W perp, or W⊥

W is a subspace of Rⁿ, its orthogonal complement, denoted ___ (read "W "), is defined as:
__ = { v ∈ Rⁿ | v · w = 0 for all w ∈ W }