Linear Algebra Practice Flashcards

A collection of vocabulary terms, definitions, and key results from Linear Algebra including matrix properties, eigenvalues, and orthogonality.

Reduced Echelon Form

A matrix configuration where it is in echelon form, the pivot of each nonzero row is $1$, and each leading $1$ is the only nonzero entry in its column.

Linear Combination

Given vectors $v_1, v_2, \text{...}, v_p$ in $\text{R}^n$ and given scalars $c_1, c_2, \text{...}, c_p$, the vector $y$ defined by $y = c_1v_1 + \text{...} + c_pv_p$ with weights $c_1, \text{...}, c_p$.

Span

The set of all linear combinations of a set of vectors $\text{\{}v_1, v_2, \text{...}, v_p\text{\}}$ denoted by $\text{Span}\text{\{}v_1, v_2, \text{...}, v_p\text{\}}$. An element $x$ belongs to the span if it can be written as $x = c_1v_1 + c_2v_2 + \text{...} + c_pv_p$.

Linear Independence

A set of vectors $\text{\{}v_1, v_2, \text{...}, v_p\text{\}}$ in $\text{R}^n$ where the homogeneous equation $x_1v_1 + x_2v_2 + \text{...} + x_pv_p = 0$ only has the trivial solution.

Matrix Transformation

A transformation $T: \text{R}^n \rightarrow \text{R}^m$ defined by the operation $T(x) = Ax$.

Jacobian

A matrix for a transformation $T$ that transforms $(u,v)$ to $(x,y)$ with $x = x(u,v)$ and $y = y(u,v)$, where the 'destination' coordinate system is in the numerator.

Eigenvalue

A scalar \text{\lambda} such that there exists a nonzero vector x \text{\in} \text{R}^n for which the equation Ax = \text{\lambda}x holds for an $n \times n$ matrix $A$.

Eigenvector

A nonzero vector $x$ as specified in the definition of an eigenvalue, such that Ax = \text{\lambda}x for some scalar \text{\lambda}.

Eigenspace ($E_{\lambda}$)

The set of all solutions of the equation Ax = \text{\lambda}x, which consists of the zero vector and all the eigenvectors corresponding to the eigenvalue \text{\lambda}.

Characteristic Equation

The equation defined by \text{det}(A - \text{\lambda}I) = 0 for a matrix $A$.

Characteristic Polynomial

The polynomial defined by the expression \text{det}(A - \text{\lambda}I).

Rank

A property of a matrix $A$, denoted by $\text{rank}(A)$, defined as the dimension of the column space of $A$.

Orthogonality (Vector to Subspace)

A vector $x$ is orthogonal to a subspace $W$ of $\text{R}^n$ if x \text{\cdot} w = 0 for each $w$ in $W$, denoted as x \text{\perp} W.

Orthogonal Complement ($W^{\perp}$)

The set of all vectors in $\text{R}^n$ that are orthogonal to all vectors in a given subspace $W$.

Orthogonal Set

A set of vectors $\text{\{}u_1, \text{...}, u_p\text{\}}$ in $\text{R}^n$ where u_i \text{\cdot} u_j = 0 for each pair with $i \neq j$.

Orthonormal Set

A set of vectors where the dot product u_i \text{\cdot} u_j = 0 if $i \neq j$ and u_i \text{\cdot} u_i = 1 if $i = j$.

Orthogonal Basis

A basis for a subspace $W$ of $\text{R}^n$ that is also an orthogonal set.

Orthonormal Basis

A basis for a subspace $W$ that is also an orthonormal set.

Gram-Schmidt Process

An algorithm to construct an orthogonal basis $\text{\{}v_1, \text{...}, v_p\text{\}}$ for a subspace $W$ starting from any basis $\text{\{}b_1, \text{...}, b_p\text{\}}$.

Similarity

Two $n \times n$ matrices $A$ and $B$ are similar if there exists an invertible $n \times n$ matrix $P$ such that $A = PBP^{-1}$.

Diagonalizable Matrix

A square matrix $A$ that is similar to a diagonal matrix $D$, meaning $A = PDP^{-1}$ for some invertible matrix $P$.

Standard Matrix of $T$

The unique $m \times n$ matrix $A$ such that $T(x) = Ax$, where the columns are the images under $T$ of the standard unit vectors: $A = [T(e_1) \text{...} T(e_n)]$.

(i,j)-cofactor

The value given by $C_{ij} = (-1)^{i+j} \text{det} A_{ij}$, where $A_{ij}$ is the matrix obtained from $A$ by deleting the $i$-th row and $j$-th column.

Coordinate Vector ($[x]_B$)

The vector of unique weights $c_1, \text{...}, c_p$ such that $x = c_1b_1 + \text{...} + c_p b_p$ for a given basis $B = \text{\{}b_1, \text{...}, b_p\text{\}}$.

Rank Theorem

The theorem stating that if a matrix $A$ has $n$ columns, then $\text{rank}(A) + \text{dim}(\text{Nul}(A)) = n$.

Orthogonal Matrix

A square matrix $U$ for which the statements $U^T U = I$, $U U^T = I$, and $U^{-1} = U^T$ are equivalent, implying its columns and rows are orthonormal.

Algebraic Multiplicity ($a.m.(\lambda_0)$)

The number of factors (\text{\lambda} - \text{\lambda}_0) in the characteristic polynomial of a matrix.

Geometric Multiplicity ($g.m.(\lambda)$)

The dimension of the eigenspace corresponding to \text{\lambda}, equal to $\text{dim}(E_{\lambda})$.