Linear Algebra and Vector Spaces Review

A collection of vocabulary flashcards covering the fundamental concepts of linear algebra, including vector spaces, linear maps, determinants, and spectral theory.

Field

A non-zero commutative ring where every non-zero element has a multiplicative inverse.

Vector Space

An abelian group $V$ with a scalar multiplication operation $F \times V \rightarrow V$ defined over a field $F$ satisfying specific distributivity and associativity axioms.

Subspace

A non-empty subset $W \subseteq V$ such that for all $u, v \in W$ and $\alpha, \beta \in F$, the linear combination $\alpha u + \beta v$ is also in $W$.

Linear Independence

A property of a vector sequence $v\_1, \dots, v\_n$ where the equation $\alpha\_1 v\_1 + \dots + \alpha\_n v\_n = 0$ implies that all scalars $\alpha\_i$ must be zero.

Basis

A vector sequence in $V$ that is both linearly independent and spans the entire vector space.

Dimension

The number of vectors in a basis of a finite-dimensional vector space, denoted as $\text{dim}(V)$.

Linear Transformation

A function $T: U \rightarrow V$ that preserves vector addition and scalar multiplication, satisfying $T(\alpha u\_1 + \beta u\_2) = \alpha T(u\_1) + \beta T(u\_2)$.

Kernel

The set of all vectors $u \in U$ such that $T(u) = 0\_V$, forming a subspace of the domain.

Image

The set of all vectors $v \in V$ such that $v = T(u)$ for some $u \in U$, forming a subspace of the codomain.

Rank

The dimension of the image of a linear transformation, denoted as $\text{rank}(T)$.

Nullity

The dimension of the kernel of a linear transformation, denoted as $\text{nullity}(T)$.

Trace

The sum of the entries on the main diagonal of a square matrix, denoted as $\text{Tr}(A)$.

Characteristic Polynomial

The determinant $\det(A - xI)$ used to determine the eigenvalues of a matrix $A$.

Eigenvalue

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

Eigenvector

A non-zero vector $v$ that, when multiplied by a matrix $A$, results in a scalar multiple of itself.

Diagonalisable

A property of a linear map or matrix that possesses a basis consisting entirely of eigenvectors.

Self-adjoint Operator

A linear operator $T$ on a Euclidean space satisfying $\tau(T(v), w) = \tau(v, T(w))$ for all vectors $v, w$.

Singular Values

The unique positive square roots of the eigenvalues of the operator $A^T A$, usually ordered as $\gamma\_1 \ge \gamma\_2 \ge \dots \ge \gamma\_n > 0$.

Proof by Induction

A method of mathematical proof where the statement is proven for a base case and then shown to hold for the next case under the assumption it holds for all previous cases.

Isomorphism

A structure-preserving mapping between two mathematical structures that allows for the comparison of their properties.

Span

The set of all possible linear combinations of a given set of vectors, representing all vectors that can be reached within a vector space.

Closure Property

A property of a set in which performing an operation on members of the set produces a member that also belongs to the same set.

Direct Sum

A way to combine two subspaces such that their intersection is the zero vector, resulting in a space that retains the properties of both subspaces.

Orthogonality

A concept in vector spaces where two vectors are orthogonal if their dot product is zero, indicating they are perpendicular.

Linear Combination

An expression formed by multiplying vectors by scalars and adding the results, used to generate new vectors in a vector space.

Eigenvalue Decomposition

A representation of a matrix into the form of its eigenvalues and eigenvectors, allowing for easier computation and analysis.

Co-dimension

The difference between the dimension of a space and the dimension of a subspace within it, indicating how 'tall' the subspace is within the larger space.

Determinants

A scalar value that can be computed from the elements of a square matrix, used to determine the matrix's invertibility and other properties.

Proof of Vector Space Axioms

To prove that a set $V$ with operations of addition and scalar multiplication forms a vector space over a field $F$, verify the following axioms: 1. Closure under addition: For all $u, v \\in V$, $u + v \\in V$. 2. Closure under scalar multiplication: For all $c \\in F$ and $u \\in V$, $c u \\in V$ . 3. Associativity of addition: For all $u, v, w \\in V$, (u + v) + w = u + (v + w)$. 4. Commutativity of addition: For all u, v \in V$,$u + v = v + u$. 5. Existence of additive identity: There exists an element$0 \in V$such that$u + 0 = u$for all$u \in V$. 6. Existence of additive inverses: For each$u \in V$, there exists$-u \in V$such that$u + (-u) = 0$. 7. Distributivity of scalar multiplication with respect to vector addition: For all$c \in F$and$u, v \in V$,$c(u + v) = cu + cv$. 8. Distributivity of scalar multiplication with respect to field addition: For all$c, d \in F$and$u \in V$,$(c + d)u = cu + du$. 9. Compatibility of scalar multiplication with field multiplication: For all$c, d \in F$and$u \in V$,$(cd)u = c(du)$. 10. Identity element of scalar multiplication: For all$u \in V$,$1u = u$$.

Basis of a Vector Space

A basis of a vector space $V$ is a set of vectors that is linearly independent and spans the entire space. To prove a set of vectors $\{v_1, v_2, \dots, v_n\}$ is a basis: 1. Show the vectors are linearly independent: The equation $\alpha_1 v_1 + \alpha_2 v_2 + \dots + \alpha_n v_n = 0$ implies that all $\alpha_i = 0$. 2. Show they span the space: Any vector $v \in V$ can be expressed as a linear combination of the basis vectors.

Linear Independence

To prove a set of vectors $\{v_1, v_2, \dots, v_n\}$ is linearly independent, show that the equation $\alpha_1 v_1 + \alpha_2 v_2 + \dots + \alpha_n v_n = 0$ implies that all coefficients $\alpha_i = 0$. This is typically done through reduction to row echelon form if placed in matrix form.

Proof of the Dimension Theorem

The dimension theorem states: If $V$ is a finite-dimensional vector space and $W$ is a subspace of $V$, then $\text{dim}(V) = \text{dim}(W) + \text{dim}(W^\perp)$, where $W^\perp$ is the orthogonal complement of $W$. Proof involves demonstrating that the basis of $V$ can be expressed in terms of bases for $W$ and its orthogonal complement.

Existence of Eigenvalues

To prove the existence of eigenvalues for a square matrix $A$, show that the characteristic polynomial $p(x) = \det(A - xI)$ has roots, which are the eigenvalues. By the Fundamental Theorem of Algebra, any polynomial of degree $n$ has $n$ roots, hence $A$ has at least one eigenvalue.

Diagonalizability Criteria

A matrix $A$ is diagonalizable if there exists a basis of eigenvectors for the space. To prove this, you show that the characteristic polynomial of $A$ has $n$ distinct roots (eigenvalues) leading to a complete set of linearly independent eigenvectors.

Trace of a Matrix

The trace of a matrix $A$ is defined as the sum of the diagonal elements, $\text{Tr}(A) = \sum_{i} a_{ii}.$ To prove its properties: 1. $\text{Tr}(AB) = \text{Tr}(BA)$ for any square matrices $A$ and $B$. 2. $\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B).$ 3. If $c$ is a scalar, then $\text{Tr}(cA) = c \text{Tr}(A).$

Properties of Determinants

Prove that the determinant of a product of matrices satisfies $\det(AB) = \det(A)\det(B)$. To prove this, you can use cofactor expansion or properties of linear transformations associated with those matrices.

Rank-Nullity Theorem

The rank-nullity theorem states: For a linear transformation $T: U \rightarrow V$, $\text{dim(kernel}(T)) + \text{dim(image}(T)) = \text{dim}(U)$. To prove this, analyze the dimension of the image and kernel using bases.

Orthogonal Complements

For any vector space $V$, the orthogonal complement $W^\perp$ is defined as the set of all vectors orthogonal to each vector in $W$. Prove that $\text{dim}(W) + \text{dim}(W^\perp) = \text{dim}(V)$ through the properties of linear transformations.

Cauchy-Schwarz Inequality

To prove the Cauchy-Schwarz Inequality, suppose $u, v$ are in an inner product space: $|\langle u, v \rangle|^2 \leq \langle u, u \rangle \langle v, v \rangle.$ This can be shown using the properties of the inner product and constructing the quadratic form associated with $t(u) + v$.

Triangle Inequality

Prove the triangle inequality for norms: $\|u + v\| \leq \|u\| + \|v\|$. Use the Cauchy-Schwarz inequality to establish relationships between the magnitudes of vector sums and their individual magnitudes.

Linear Transformation Properties

To show a function $T: U \rightarrow V$ is a linear transformation, verify two properties: 1. $T(u + v) = T(u) + T(v)$ for all $u, v \in U$. 2. $T(cu) = cT(u)$ for all $c \in F$ and $u \in U$.

Subspace Criteria

To prove that a set $W$ is a subspace of a vector space $V$, verify that: 1. $W$ is non-empty. 2. If $u, v \in W$, then $u + v \in W$. 3. If $u \in W$ and $c \in F$, then $cu \in W$.

Proof of Existence of Solutions

To prove the existence of solutions for the equation $Ax = b$ (where $A$ is a matrix), establish that the rank of $A$ is equal to the rank of the augmented matrix $[A|b]$, which guarantees at least one solution exists.

Eigenvalues and Eigenvectors

For a matrix $A$, to prove that a non-zero vector $v$ is an eigenvector corresponding to an eigenvalue $\lambda$, show that the equation $Av = \lambda v$ holds. This entails solving the characteristic equation.

Identity Matrix Properties

The identity matrix $I$ has the property that for any matrix $A$, $AI = A$ and $IA = A$. To prove this, use the definition of matrix multiplication.

Linear Combinations

Prove that any vector in the vector space can be expressed as a linear combination of basis vectors. This requires defining the basis and showing the representation of any vector using these basis vectors.

Direct Sum of Subspaces

To prove that two subspaces $U$ and $W$ of a vector space $V$ form a direct sum $U \oplus W$, show that their intersection is only the zero vector: $U \cap W = \{0\}$ and every element of $V$ can be written uniquely as the sum of an element from $U$ and an element from $$W$.

Orthogonal Sets of Vectors

To prove that a set of vectors is orthogonal, show that the dot product of every pair of distinct vectors within the set equals zero: $\langle u_i, u_j \rangle = 0$ for all $i \neq j$.

Convergence of Sequences in Metric Spaces

To prove that a sequence converges in a metric space, show that for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \\geq N$, $d(x_n, L) < \epsilon$, where $L$ is the limit.

Continuity of Linear Maps

To prove that a linear map is continuous, show that for a linear transformation $T$, for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $\|u - v\| < \delta$ then $\|T(u) - T(v)\| < \epsilon$.

Bounded Linear Operators

To prove that a linear operator $T$ is bounded, show that there exists a constant $C\ge0$ such that for all vectors $u$ in the domain, $\|T(u)\| \leq C\|u\|$.

Change of Basis Formula

To prove the change of basis formula, if $[x]_{B}$ is the coordinate vector of $x$ in basis $B$ and $[x]_{C}$ in basis $C$, then $[x]_{C} = P_{C \leftarrow B}[x]_{B}$ where $P_{C \leftarrow B}$ is the change of basis matrix.

Application of the Intermediate Value Theorem

To prove the Intermediate Value Theorem, show that if $f$ is continuous on $[a, b]$ and $N$ is a number between $f(a)$ and $f(b)$, then there exists some $c \in (a, b)$ such that $f(c) = N$.

Fundamental Theorem of Algebra

To prove the Fundamental Theorem of Algebra, show that any non-constant polynomial $p(z)$ has at least one complex root. This is often done using properties of complex numbers and topological arguments.

Linear Functionals

A linear functional is a mapping from a vector space to its field. To prove it is linear, you must show $f(u + v) = f(u) + f(v)$ and $f(cu) = c f(u)$ for all $u, v$ in the vector space and all scalars $c$.

Scalar Product and Cauchy-Schwarz Inequality

In the context of scalar products, the Cauchy-Schwarz inequality can be formally proved by establishing that $|\langle u, v \rangle|^2 \leq \|u\|^2 \|v\|^2$ through applying concepts of inner product spaces.

Norm of a Vector Space

To prove a function $\|v\|$ is a norm on vector space $V$, verify: 1. $\|v\| \geq 0$, and $\|v\| = 0$ if and only if $v = 0$. 2. \|\alpha v\| = |
|\|v\| for all $\alpha \in F$ and $v \in V$. 3. $\|u + v\| \leq \|u\| + \|v\|$ (Triangle Inequality).

Rank of a Matrix

The rank of a matrix is the dimension of its row space or column space. To prove this equality, transform the matrix to reduced row echelon form and count the non-zero rows, demonstrating that this count equals the maximum number of linearly independent rows or columns.

Fixed Point Theorem

To prove a fixed point theorem (like Banach’s), demonstrate that a contraction mapping in a complete metric space has a unique fixed point using the properties of metric spaces and sequences.

Laplacian Matrix and Graph Theory

To prove the properties of the Laplacian matrix in graph theory, confirm that the sum of each row (or column) equals zero, illustrating the relationship between the graph’s structure and its Laplacian representation.

Gram-Schmidt Process

To prove that the Gram-Schmidt process produces an orthogonal (or orthonormal) set from a basis, show that the successive steps accurately compute the orthogonal projections of vectors and form the required orthogonality.

De Moivre's Theorem

To prove De Moivre's theorem, $\text{cis}(\theta)^n = \text{cis}(n\theta)$, utilize the properties of Euler's formula and induction to establish the equalities.

Claim about Linear Combinations

To prove that a specific combination of linear combinations is an element of a vector space, show that the linear combination can be expressed in terms of basis vectors, maintaining closure under vector operations.

Properties of Complex Eigenvalues

To prove that complex eigenvalues of a real matrix come in conjugate pairs, utilize the characteristic polynomial properties and the fact that real matrices’ eigenvalue roots are either real or occur in conjugate pairs.

Independence of Eigenvectors

To show that eigenvectors associated with distinct eigenvalues are linearly independent, assume a linear combination equals zero, leading to a contradiction based on the eigenvalue definition.

Dual Spaces

The dual space of a vector space $V$ consists of all linear functionals on $V$. To prove properties about dual spaces, verify that the dimension of $V^{*}$ is equal to the dimension of $V$.

Implication of Zero Divisors in Rings

Prove that if a ring has nontrivial zero divisors, then the product of these zero divisors produces a zero, demonstrating structural properties about the ring.

Closure Under Addition of Vectors

Prove the closure property $(u + v) \in V$ for all $u, v \in V$ within a specified vector space by showing that the sum of any two vectors results in another vector that adheres to the space’s defined dimension.

Application of the Matrix Exponential

To prove the properties of matrix exponentials, confirm that if $A$ is a matrix, then $e^{A} = \sum_{n=0}^{\infty} \frac{A^n}{n!}$ converges and can be applied in solving linear differential equations.