Linear Algebra

Matrix

A rectangular array of numbers arranged in rows and columns.

(m \times n) matrix

A matrix with (m) rows and (n) columns.

Row

A horizontal line of entries in a matrix.

Column

A vertical line of entries in a matrix.

Entry

An individual number in a matrix, often written (a_{ij}), where (i) identifies the row and (j) identifies the column.

Row index

The first subscript (i) in (a_{ij}); it identifies an entry’s row.

Column index

The second subscript (j) in (a_{ij}); it identifies an entry’s column.

Square matrix

A matrix whose number of rows equals its number of columns; equivalently, an (m \times n) matrix with (m=n).

Diagonal of a matrix

The entries (a_{ij}) for which (i=j), running from the upper-left toward the lower-right.

Upper triangular matrix

A matrix in which every entry below the diagonal is zero; equivalently, (a_{ij}=0) whenever (i>j).

Lower triangular matrix

A matrix in which every entry above the diagonal is zero; equivalently, (a_{ij}=0) whenever (i<j).

Diagonal matrix

A matrix in which every off-diagonal entry is zero; equivalently, (a_{ij}=0) whenever (i\ne j).

Off-diagonal entry

Any matrix entry not on the diagonal—that is, an entry (a_{ij}) where (i\ne j).

(\operatorname{diag}(d_1,\ldots,d_n))

Notation for the (n\times n) diagonal matrix with (d_1,\ldots,d_n) as its diagonal entries.

Omitted entry

A blank matrix entry that is understood to be zero unless stated otherwise.

Vector

An ordered list of numbers, often treated as a one-column matrix.

(\mathbb{R}^n)

The set of all vectors with (n) real-number entries.

Zero vector

A vector in which every coordinate is zero.

Zero matrix

A matrix in which every entry is zero.

Zero object notation

The notation (\mathbf{0}), often with a subscript such as (\mathbf{0}n) or (\mathbf{0}{m\times n}), indicating a zero vector or zero matrix of a particular size.

Nonzero vector

A vector with at least one entry that is not zero.

Nonzero matrix

A matrix with at least one entry that is not zero.

Identity matrix

The (n\times n) diagonal matrix with 1s on the diagonal and 0s everywhere else.

(I_n)

Notation for the (n\times n) identity matrix.

Standard basis vectors

The columns of (I_n), denoted (\mathbf e_1,\ldots,\mathbf e_n). Each has exactly one entry equal to 1 and all remaining entries equal to 0.

(\mathbf e_i)

The (i)-th standard basis vector: the vector with a 1 in position (i) and zeros elsewhere.

First pivot column

The first column of a matrix that is nonzero, meaning it has at least one nonzero entry.

Scalar

A single number used to multiply a vector or matrix.

Scalar multiple

A vector obtained by multiplying every entry of another vector by the same scalar.

Rank-one matrix

A matrix in which every column is a scalar multiple of its first pivot column.

Rank

A matrix concept related to the number of independent directions represented by its columns.

Symmetric matrix

A square matrix equal to its transpose, so (a_{ij}=a_{ji}).

Computer algebra system

Software that can perform symbolic and numerical mathematical computations.

Sage

A mathematical software system used to inspect matrix properties and construct matrices.

Matrix constructor

A built-in software command for creating a particular kind of matrix, such as a zero matrix, identity matrix, or diagonal matrix.

Scalar

A real number; scalars are the basic numerical objects used to build vectors and matrices.

Real numbers \(\mathbb{R}\)

All numbers on the number line, including negative numbers, zero, fractions, and irrational numbers such as \(\pi\).

Membership symbol \(\in\)

A symbol meaning “is an element of” or “belongs to.”

Vector

A vertical list of \(n\) scalars; a vector in \(\mathbb{R}^n\).

\(\mathbb{R}^n\)

The set of all vectors with \(n\) real-number coordinates.

Coordinate

One of the scalar entries in a vector.

Matrix

An array of scalars organized into rows and columns.

\(m \times n\) matrix

A matrix with \(m\) rows and \(n\) columns.

Row

A horizontal list of entries in a matrix.

Column

A vertical list of entries in a matrix.

Matrix entry

An individual scalar in a matrix.

\(i,j\) entry

The entry located in the \(i\)-th row and \(j\)-th column of a matrix, written \(a_{ij}\).

Scalar-vector product

Multiplying every coordinate of a vector by the same scalar.

Vector addition

Adding two vectors coordinate by coordinate; the vectors must have the same number of coordinates.

Scalar-matrix product

Multiplying every entry of a matrix by the same scalar.

Matrix addition

Adding two matrices entry by entry; the matrices must have the same number of rows and columns.

Shape

The dimensions of a vector or matrix. Vectors must have the same number of coordinates, and matrices must have the same dimensions, to be added.

Transpose

The matrix formed by interchanging the rows and columns of another matrix, written with a superscript \(T\).

Transpose notation

For a matrix \(A\), its transpose is written \(A^T\).

Transpose entry rule

The \(i,j\) entry of \(A^T\) equals the \(j,i\) entry of \(A\).

Transpose dimensions

If \(A\) is an \(m \times n\) matrix, then \(A^T\) is an \(n \times m\) matrix.

Horizontal vector notation

Writing a column vector as a row and applying a transpose symbol to save vertical space.

Linear operation

An operation that distributes over addition and scalar multiplication.

Linearity of transpose

\((c_1A_1+c_2A_2)^T=c_1A_1^T+c_2A_2^T\).

Involution

An operation that returns the original object when performed twice.

Involutive operation of transpose

\((A^T)^T=A\).

Symmetric matrix

A square matrix equal to its own transpose: \(S^T=S\).

Diagonal of a matrix

The entries \(a_{ii}\) running from the upper-left corner to the lower-right corner of a square matrix.

Symmetry across the diagonal

The property of a symmetric matrix in which \(a_{ij}=a_{ji}\); entries mirror one another across the diagonal.

Trace

The sum of the diagonal entries of an \(n \times n\) matrix.

Trace notation

The trace of a matrix \(A\) is often written \(\operatorname{tr}(A)\).

Trace as a scalar

The trace of a square matrix is a single number, not a matrix.

Linearity of trace

\(\operatorname{tr}(c_1A_1+c_2A_2)=c_1\operatorname{tr}(A_1)+c_2\operatorname{tr}(A_2)\).

Computer algebra system

Software that performs mathematical calculations symbolically or numerically.

Sage

A Python-based computer algebra system used in the lesson for calculations with scalars, vectors, and matrices.

Matrix syntax in Sage

The `matrix(...)` command used to define a matrix by listing its rows.

Vector syntax in Sage

The `vector(...)` command used to define a vector by listing its coordinates.

Matrix transpose syntax in Sage

The `.T` notation applied to a matrix to obtain its transpose.

Matrix trace syntax in Sage

The `.trace()` command applied to a square matrix to calculate its trace. [youtube](https://www.youtube.com/watch?v=3uNbT8X7zGo)

Linear combination

An expression formed by multiplying vectors by scalars and adding the results, such as \(c_1\mathbf v_1+\cdots+c_m\mathbf v_m\).

Weight

A scalar used to multiply a vector in a linear combination.

Matrix-vector product

The product \(A\mathbf v\), defined as a linear combination of the columns of \(A\), with the coordinates of \(\mathbf v\) as weights.

Compatibility criterion

The requirement that the number of coordinates in the input vector equals the number of columns of the matrix before \(A\mathbf v\) is defined.

Compatible matrix-vector product

A valid product \(A\mathbf v\) in which \(A\) has the same number of columns as \(\mathbf v\) has coordinates.

Illegal matrix-vector product

An undefined expression in which the vector’s number of coordinates does not equal the matrix’s number of columns.

\(m \times n\) matrix

A matrix with \(m\) rows and \(n\) columns.

Input vector

For an \(m\times n\) matrix \(A\), a vector \(\mathbf v\in\mathbb R^n\) that can be multiplied by \(A\).

Output vector

The result \(A\mathbf v\); if \(A\) is \(m\times n\), the output belongs to \(\mathbb R^m\).

Column combination

A linear combination whose vectors are the columns of a matrix.

Column weights

The coordinates of the vector used in a matrix-vector product; each coordinate scales the corresponding matrix column.

Matrix as a transformation

The view of a matrix as a machine that maps input vectors to output vectors through matrix-vector multiplication.

Matrix-vector product dimensions

If \(A\) is an \(m\times n\) matrix and \(\mathbf v\in\mathbb R^n\), then \(A\mathbf v\in\mathbb R^m\).

Linearity of matrix-vector multiplication

The rule \(A(c_1\mathbf v_1+c_2\mathbf v_2)=c_1A\mathbf v_1+c_2A\mathbf v_2\).

Identity rule

The rule \(I_n\mathbf v=\mathbf v\) for every vector \(\mathbf v\in\mathbb R^n\).

Identity matrix \(I_n\)

The \(n\times n\) matrix with 1s on the diagonal and 0s elsewhere; it leaves compatible vectors unchanged when multiplied by them.

Zero vector

The vector whose coordinates are all zero.

Zero rule

The rule \(A\mathbf0=\mathbf0\) for any matrix \(A\) with a compatible zero vector.

Arrow notation

A notation describing a matrix as a mapping from one vector space to another, such as \(A:\mathbb R^n\to\mathbb R^m\).

\(A:\mathbb R^n\to\mathbb R^m\)

Arrow notation stating that an \(m\times n\) matrix \(A\) takes vectors in \(\mathbb R^n\) as inputs and produces vectors in \(\mathbb R^m\) as outputs.

Square matrix

A matrix with the same number of rows and columns.

Eigenvector

A nonzero vector \(\mathbf v\) for which multiplying by a matrix \(A\) only scales the vector: \(A\mathbf v=\lambda\mathbf v\).

Eigenvalue

The scalar \(\lambda\) associated with an eigenvector in the equation \(A\mathbf v=\lambda\mathbf v\).

Eigenvector equation

The equation \(A\mathbf v=\lambda\mathbf v\), which defines an eigenvector and its associated eigenvalue.

Associated eigenvalue

The eigenvalue paired with a particular eigenvector of a matrix.

Eigenspace notation

Notation such as \(\mathbf v\in E_A(\lambda)\), meaning that \(\mathbf v\) satisfies \(A\mathbf v=\lambda\mathbf v\).