Algebra and Linear Algebra Flashcards

This vocabulary flashcard set covers essential definitions from the lecture including vector spaces, subspaces, basis properties, linear transformations, matrices, determinants, and spectral theory.

Field

A non-zero commutative ring in which every non-zero element has a multiplicative inverse, such as the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$.

Vector Space

An abelian group $V$ with a scalar multiplication operation from a field $F$ that follows specific distributive and associative axioms.

Subspace

A non-empty subset of a vector space that is closed under addition and scalar multiplication, making it a vector space itself.

Linear Combination

An expression of the form $\alpha_1 v_1 + \dots + \alpha_n v_n$ where scalars multiply a sequence of vectors.

Linearly Independent

A property of a vector sequence where the only way to obtain the zero vector through a linear combination is for all scalar coefficients to be zero.

Span

The set of all possible linear combinations that can be formed from a finite sequence of vectors.

Basis

A vector sequence that is both linearly independent and spans the vector space; every vector in the space is uniquely representable through it.

Dimension

The number of vectors contained in a basis of a finite-dimensional vector space, denoted as $dim(V)$. Mexico

Linear Transformation

A mapping between vector spaces $T: U \rightarrow V$ that preserves vector addition and scalar multiplication.

Kernel

The set of all vectors in the domain that map to the zero vector in the codomain, also called the nullspace.

Image

The set of all vectors in the codomain that are mapped to by elements of the domain, also called the range.

Rank

The dimension of the image of a linear transformation or matrix.

Nullity

The dimension of the kernel of a linear transformation or matrix.

Rank-Nullity Formula

The theorem stating that for a linear map $T: U \rightarrow V$, $rank(T) + nullity(T) = dim(U)$.

Direct Sum

A decomposition of a vector space into two subspaces $W_1 \oplus W_2$ where their intersection is trivial and their sum covers the whole space.

Euclidean Space

A real vector space equipped with a symmetric, bilinear, and positive-definite Euclidean form.

Orthonormal Basis

A basis where all vectors have unit length and are mutually orthogonal to each other.

Change of Basis Matrix

An invertible matrix $P$ used to convert coordinate vectors from an old basis to a new basis.

Similar Matrices

Matrices $A$ and $B$ such that $B = PAP^{-1}$ for an invertible matrix $P$, representing the same operator in different bases.

Smith Normal Form

A canonical diagonal-like form of a matrix produced through elementary row and column operations to represent a linear mapping.

Row Reduced Echelon Form

A matrix state obtained via row operations where leading entries are 1 and are the sole non-zero entries in their columns.

Determinant

A scalar quantity associated with a square matrix that determines invertibility and identifies volume scaling factors.

Characteristic Polynomial

The polynomial $det(A - xI)$ whose roots identify the eigenvalues of a matrix $A$.

Eigenvalue

A scalar $\lambda$ for which there exists a non-zero vector $v$ such that $T(v) = \lambda v$.

Trace

The sum of the diagonal elements of a square matrix, which is invariant under similarity transformations.

Spectral Theorem

The theorem stating that every self-adjoint operator on a Euclidean space has an orthonormal basis composed of eigenvectors.

Singular Values

The unique positive values $\gamma_1, \dots, \gamma_n$ that appear in the diagonal of a matrix's singular value decomposition.

Field Axiom A1

Commutativity of Addition: $\forall \\alpha, \\beta \\in F, \\alpha + \\beta = \\beta + \\alpha$

Field Axiom A2

Associativity of Addition: $\forall \\alpha, \\beta, \\gamma \\in F, (\\alpha + \\beta) + \\gamma = \\alpha + (\\beta + \\gamma)$

Field Axiom A3

Additive Unity: There exists $0 \\in F$ such that $\alpha + 0 = 0 + \alpha = \alpha$

Field Axiom A4

Additive Inverse: $\\forall \\alpha \\in F, \\\text{there exists} \\ -\\alpha \\in F \\text{such that} \\alpha + (−\\alpha) = (−\\alpha) + \alpha = 0$

Field Axiom M1

Commutativity of Multiplication: $\forall \\alpha, \\beta \\in F, \\alpha \\beta = \\beta \\alpha$

Field Axiom M2

Associativity of Multiplication: $\forall \\alpha, \\beta, \\gamma \\in F, (\\alpha \\beta)\\gamma = \\alpha(\\beta\\gamma)$

Field Axiom M3

Multiplicative Unity: There exists $1 \\in F$ such that $\\alpha \cdot 1 = 1 \cdot \alpha = \alpha$

Field Axiom M4

Multiplicative Inverse: For each $\\alpha \\in F$ with $\\alpha \neq 0$, there exists $\\alpha^{-1} \\in F$ such that $\\alpha \\cdot \\alpha^{-1} = \\alpha^{-1} \\cdot \\alpha = 1$

Distributive Axiom D

$$\\forall \\alpha, \\beta, \\gamma \\in F, (\\alpha + \\beta)\\gamma = \\alpha\\gamma + \\beta\\gamma$$

Vector Space Definition

A vector space is an abelian group V with scalar multiplication that satisfies certain axioms: $\alpha(u + v) = \alpha u + \alpha v$, $(\alpha + \beta)v = \alpha v + \beta v$, $(\alpha \beta)v = \alpha(\beta v)$, and $1v = v$.

Linear Dependence Definition

A vector sequence $v_1, v_2, \\dots, v_n$ is linearly dependent if there exist scalars $\alpha_1, \alpha_2, \\dots, \alpha_n$ (not all zero) such that $\alpha_1v_1 + \alpha_2v_2 + \\dots + \alpha_nv_n = 0$.

Existence of Basis Theorem

If a vector sequence $v_1, v_2, \\dots, v_r$ spans V, then there

Linear Combination Theorem

A vector $v$ in a vector space can be represented as a linear combination of basis vectors of the space.

Basis of a Vector Space Theorem

A vector space can have multiple bases, all of which contain the same number of elements, which is the dimension of the space.

Dimension Theorem

The dimension of a vector space is the maximum number of linearly independent vectors in that space.

Direct Sum Theorem

If a vector space V can be expressed as the direct sum of subspaces $W_1$ and $W_2$, then every vector in V can be uniquely expressed as a sum of vectors from each subspace.

Rank-Nullity Theorem

For a linear transformation $T: U \rightarrow V$, the equation $rank(T) + nullity(T) = dim(U)$ holds true.

Eigenvector Definition

A non-zero vector $v$ such that when a linear transformation $T$ is applied to it, the output is a scalar multiple of $v$, i.e., $T(v) = \lambda v$.

Eigenvalue Theorem

The scalar $\lambda$ is an eigenvalue of a matrix A if there exists a non-zero vector $v$ such that $Av = \lambda v$.

Minimal Polynomial Theorem

The minimal polynomial of a matrix provides the smallest polynomial for which the matrix is a root, affecting its eigenvalues and eigenvectors.

Characteristic Polynomial Theorem

The roots of the characteristic polynomial $det(A - xI)$ are the eigenvalues of the matrix A.

Orthonormal Basis Definition

An orthonormal basis consists of vectors that are all orthogonal to each other and each have unit length.

Linear Transformation Definition

A mapping between vector spaces $T: U \rightarrow V$ that preserves vector addition and scalar multiplication.

Kernel Definition

The set of all vectors in the domain that map to the zero vector in the codomain, also called the nullspace.

Image Definition

The set of all vectors in the codomain that are mapped to by elements of the domain, also called the range.

Rank Definition

The dimension of the image of a linear transformation or matrix.

Nullity Definition

The dimension of the kernel of a linear transformation or matrix.

Rank-Nullity Formula

The theorem stating that for a linear map $T: U \rightarrow V$, $rank(T) + nullity(T) = dim(U)$.

Direct Sum Definition

A decomposition of a vector space into two subspaces $W\_1 \oplus W\_2$ where their intersection is trivial and their sum covers the whole space.

Euclidean Space Definition

A real vector space equipped with a symmetric, bilinear, and positive-definite Euclidean form.

Orthonormal Basis Properties

A basis where all vectors have unit length and are mutually orthogonal to each other.

Change of Basis Matrix

An invertible matrix $P$ used to convert coordinate vectors from an old basis to a new basis.