Definition (Euclidean Space \(\mathbb{R}^n\)). For any positive integer \(n\), the set of all elements of the form \((x_1, \ldots, x_n)\) where \(x_i \in \mathbb{R}\) for \(1 \leq i \leq n\) is called n-dimensional Euclidean space and is denoted \(\mathbb{R}^n\). The elements of \(\mathbb{R}^n\) are called points.

Definition (Linear Combination). Let \(\vec{v}_1, \ldots, \vec{v}_k \in \mathbb{R}^n\). The sum \(c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_k\vec{v}_k\) where \(c_i \in \mathbb{R}\) for \(1 \leq i \leq k\) is called a linear combination of \(\vec{v}_1, \ldots, \vec{v}_k\).

Definition (Span). Let \(B = \{\vec{v}_1, \ldots, \vec{v}_k\}\) be a set of vectors in \(\mathbb{R}^n\). Then we define \(\operatorname{Span} B = \{c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_k\vec{v}_k \mid c_1, \ldots, c_k \in \mathbb{R}\}\). We say that \(\operatorname{Span} B\) is spanned by \(B\) and that \(B\) is a spanning set for \(\operatorname{Span} B\).

Definition (Vector Equation). If the set \(S\) is spanned by \(\{\vec{v}_1, \ldots, \vec{v}_k\}\), then a vector equation for \(S\) is \(\vec{x} = c_1\vec{v}_1 + \cdots + c_k\vec{v}_k\), \(c_1, \ldots, c_k \in \mathbb{R}\).

Definition (Linearly Dependent / Independent). A set of vectors \(\{\vec{v}_1, \ldots, \vec{v}_k\}\) in \(\mathbb{R}^n\) is linearly dependent if there exist coefficients \(c_1, \ldots, c_k\), not all zero, such that \(\vec{0} = c_1\vec{v}_1 + \cdots + c_k\vec{v}_k\). The set is linearly independent if the only solution is \(c_1 = c_2 = \cdots = c_k = 0\) (the trivial solution).

Definition (Basis). If a subset \(S\) of \(\mathbb{R}^n\) can be written as a span of vectors \(\vec{v}_1, \ldots, \vec{v}_k\) where \(\{\vec{v}_1, \ldots, \vec{v}_k\}\) is linearly independent, then \(\{\vec{v}_1, \ldots, \vec{v}_k\}\) is a basis for \(S\).

Definition (Standard Basis). In \(\mathbb{R}^n\), let \(\vec{e}_i\) represent the vector whose \(i\)-th component is 1 and all other components are 0. The set \(\{\vec{e}_1, \ldots, \vec{e}_n\}\) is called the standard basis for \(\mathbb{R}^n\).

Definition (Line in \(\mathbb{R}^n\)). Let \(\vec{v}, \vec{b} \in \mathbb{R}^n\) with \(\vec{v} \neq \vec{0}\). The set with vector equation \(\vec{x} = c_1\vec{v} + \vec{b}\), \(c_1 \in \mathbb{R}\), is a line in \(\mathbb{R}^n\) passing through \(\vec{b}\).

Definition (Plane in \(\mathbb{R}^n\)). Let \(\vec{v}_1, \vec{v}_2, \vec{b} \in \mathbb{R}^n\) with \(\{\vec{v}_1, \vec{v}_2\}\) linearly independent. The set with vector equation \(\vec{x} = c_1\vec{v}_1 + c_2\vec{v}_2 + \vec{b}\), \(c_1, c_2 \in \mathbb{R}\), is a plane in \(\mathbb{R}^n\) passing through \(\vec{b}\).

Definition (\(k\)-plane in \(\mathbb{R}^n\)). Let \(\vec{v}_1, \ldots, \vec{v}_k, \vec{b} \in \mathbb{R}^n\) with \(\{\vec{v}_1, \ldots, \vec{v}_k\}\) linearly independent. The set with vector equation \(\vec{x} = c_1\vec{v}_1 + \cdots + c_k\vec{v}_k + \vec{b}\), \(c_1, \ldots, c_k \in \mathbb{R}\), is a \(k\)-plane in \(\mathbb{R}^n\) passing through \(\vec{b}\).

Definition (Hyperplane in \(\mathbb{R}^n\)). Let \(\vec{v}_1, \ldots, \vec{v}_{n-1}, \vec{b} \in \mathbb{R}^n\) with \(\{\vec{v}_1, \ldots, \vec{v}_{n-1}\}\) linearly independent. The set with vector equation \(\vec{x} = c_1\vec{v}_1 + \cdots + c_{n-1}\vec{v}_{n-1} + \vec{b}\), \(c_i \in \mathbb{R}\), is a hyperplane in \(\mathbb{R}^n\) passing through \(\vec{b}\).

Definition (Subspace of \(\mathbb{R}^n\)). A non-empty subset \(S\) of \(\mathbb{R}^n\) is called a subspace of \(\mathbb{R}^n\) if it satisfies all ten properties V1–V10 of Theorem 1.1.1.

Definition (Dot Product). Let \(\vec{x}, \vec{y} \in \mathbb{R}^n\). The dot product of \(\vec{x}\) and \(\vec{y}\) is \(\vec{x} \cdot \vec{y} = x_1 y_1 + \cdots + x_n y_n = \sum_{i=1}^n x_i y_i\).

Definition (Length / Norm). Let \(\vec{x} \in \mathbb{R}^n\). The length (or norm) of \(\vec{x}\) is \(\|\vec{x}\| = \sqrt{\vec{x} \cdot \vec{x}}\).

Definition (Unit Vector). A vector \(\vec{x} \in \mathbb{R}^n\) such that \(\|\vec{x}\| = 1\) is called a unit vector.

Definition (Angle in \(\mathbb{R}^n\)). Let \(\vec{x}, \vec{y} \in \mathbb{R}^n\). The angle between \(\vec{x}\) and \(\vec{y}\) is the angle \(\theta\) such that \(\vec{x} \cdot \vec{y} = \|\vec{x}\|\|\vec{y}\| \cos \theta\).

Definition (Orthogonal). Two vectors \(\vec{x}, \vec{y} \in \mathbb{R}^n\) are orthogonal if and only if \(\vec{x} \cdot \vec{y} = 0\).

Definition (Cross Product). Let \(\vec{v}, \vec{w} \in \mathbb{R}^3\). The cross product is \(\vec{v} \times \vec{w} = \begin{pmatrix} v_2 w_3 - v_3 w_2 \\ v_3 w_1 - v_1 w_3 \\ v_1 w_2 - v_2 w_1 \end{pmatrix}\).

Definition (Projection). Let \(\vec{u}, \vec{v} \in \mathbb{R}^n\) with \(\vec{v} \neq \vec{0}\). The projection of \(\vec{u}\) onto \(\vec{v}\) is \(\operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2} \vec{v}\).

Definition (Perpendicular). Let \(\vec{u}, \vec{v} \in \mathbb{R}^n\) with \(\vec{v} \neq \vec{0}\). The perpendicular of \(\vec{u}\) onto \(\vec{v}\) is \(\operatorname{perp}_{\vec{v}} \vec{u} = \vec{u} - \operatorname{proj}_{\vec{v}} \vec{u}\).

Definition (Linear Equation). An equation of the form \(a_1 x_1 + \cdots + a_n x_n = b\) where \(a_1, \ldots, a_n, b\) are constants is called a linear equation. The constants \(a_i\) are the coefficients and \(b\) is the right-hand side.

Definition (System of Linear Equations). A set of \(m\) linear equations in the same variables \(x_1, \ldots, x_n\) is called a system of linear equations.

Definition (Solution). A vector \(\vec{s} \in \mathbb{R}^n\) is a solution of a system of \(m\) linear equations in \(n\) unknowns if all \(m\) equations are satisfied when \(x_i = s_i\) for \(1 \leq i \leq n\).

Definition (Consistent / Inconsistent). A system is consistent if it has at least one solution; otherwise it is inconsistent.

Definition (Solution Set). The set of all solutions of a system of linear equations is called the solution set of the system.

Definition (Augmented Matrix / Coefficient Matrix). The augmented matrix of a system is the rectangular array \([A \mid \vec{b}]\) formed by the coefficients and the right-hand side. The coefficient matrix \(A\) contains only the coefficients.

Definition (Equivalent Systems). Two systems of equations are equivalent if they have the same solution set.

Definition (Elementary Row Operations). The three elementary row operations (EROs) are:

1. Multiplying a row by a non-zero scalar (\(cR_i\)).
   
2. Adding a multiple of one row to another (\(R_i + cR_j\)).
   
3. Swapping two rows (\(R_i \leftrightarrow R_j\)).

Definition (Row Equivalent). Two matrices \(A\) and \(B\) are row equivalent if there exists a sequence of EROs transforming \(A\) into \(B\).

Definition (Reduced Row Echelon Form). A matrix \(R\) is in RREF if:

1. All rows with a non-zero entry are above rows of all zeros.
   
2. The first non-zero entry in each non-zero row is 1 (a leading one).
   
3. Each leading one is to the right of the leading one in any row above it.
   
4. A leading one is the only non-zero entry in its column.

Definition (Free Variable). Let \(R\) be the RREF of the coefficient matrix. If the \(j\)-th column of \(R\) does not contain a leading one, then \(x_j\) is a free variable.

Definition (Rank). The rank of a matrix is the number of leading ones in its RREF.

Definition (Homogeneous System). A system of linear equations is homogeneous if the right-hand side is all zeros, i.e., it has the form \([A \mid \vec{0}]\).

Definition (Matrix). An \(m \times n\) matrix \(A\) is a rectangular array with \(m\) rows and \(n\) columns. We denote the entry in the \(i\)-th row and \(j\)-th column by \(a_{ij}\). The set of all \(m \times n\) matrices with real entries is denoted \(M_{m \times n}(\mathbb{R})\).

Definition (Matrix Addition and Scalar Multiplication). Let \(A, B \in M_{m \times n}(\mathbb{R})\) and \(c \in \mathbb{R}\). Then \((A + B)_{ij} = a_{ij} + b_{ij}\) and \((cA)_{ij} = c \cdot a_{ij}\).

Definition (Transpose). The transpose of an \(m \times n\) matrix \(A\) is the \(n \times m\) matrix \(A^T\) with \((A^T)_{ij} = a_{ji}\).

Definition (Matrix-Vector Multiplication, Row View). Let \(A\) be an \(m \times n\) matrix with rows \(\vec{a}_i^T\). For \(\vec{x} \in \mathbb{R}^n\), \(A\vec{x} = \begin{pmatrix} \vec{a}_1 \cdot \vec{x} \\ \vdots \\ \vec{a}_m \cdot \vec{x} \end{pmatrix}\).

Definition (Matrix-Vector Multiplication, Column View). Let \(A\) be an \(m \times n\) matrix with columns \(\vec{a}_1, \ldots, \vec{a}_n\). For \(\vec{x} \in \mathbb{R}^n\), \(A\vec{x} = x_1\vec{a}_1 + \cdots + x_n\vec{a}_n\).

Definition (Matrix Multiplication). Let \(A\) be \(m \times n\) and \(B = [\vec{b}_1 \cdots \vec{b}_p]\) be \(n \times p\). Then \(AB = [A\vec{b}_1 \cdots A\vec{b}_p]\), an \(m \times p\) matrix with \((AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}\).

Definition (Identity Matrix). The \(n \times n\) identity matrix \(I_n\) has \((I)_{ii} = 1\) and \((I)_{ij} = 0\) for \(i \neq j\). Its columns are the standard basis vectors of \(\mathbb{R}^n\).

Definition (Block Matrix). An \(m \times n\) matrix \(A\) can be written as a \(k \times \ell\) block matrix where \(A_{ij}\) are submatrices (blocks) such that blocks in the same row have the same number of rows and blocks in the same column have the same number of columns. Block multiplication follows the same formula as ordinary matrix multiplication.

Definition (Linear Mapping). A function \(L : \mathbb{R}^n \to \mathbb{R}^m\) is a linear mapping if for every \(\vec{x}, \vec{y} \in \mathbb{R}^n\) and \(b, c \in \mathbb{R}\), \(L(b\vec{x} + c\vec{y}) = bL(\vec{x}) + cL(\vec{y})\).

Definition (Standard Matrix). The matrix \([L] = [L(\vec{e}_1) \cdots L(\vec{e}_n)]\) is the standard matrix of the linear mapping \(L\).

Definition (Kernel). Let \(L : \mathbb{R}^n \to \mathbb{R}^m\) be linear. The kernel of \(L\) is \(\ker(L) = \{\vec{x} \in \mathbb{R}^n \mid L(\vec{x}) = \vec{0}\}\).

Definition (Range). Let \(L : \mathbb{R}^n \to \mathbb{R}^m\) be linear. The range of \(L\) is \(R(L) = \{L(\vec{x}) \in \mathbb{R}^m \mid \vec{x} \in \mathbb{R}^n\}\).

Definition (Nullspace). Let \(A\) be \(m \times n\). The nullspace of \(A\) is \(\operatorname{Null}(A) = \{\vec{x} \in \mathbb{R}^n \mid A\vec{x} = \vec{0}\}\).

Definition (Columnspace). Let \(A = [\vec{a}_1 \cdots \vec{a}_n]\). The columnspace is \(\operatorname{Col}(A) = \operatorname{Span}\{\vec{a}_1, \ldots, \vec{a}_n\} = \{A\vec{x} \in \mathbb{R}^m \mid \vec{x} \in \mathbb{R}^n\}\).

Definition (Rowspace). The rowspace of an \(m \times n\) matrix \(A\) is \(\operatorname{Row}(A) = \{A^T \vec{x} \in \mathbb{R}^n \mid \vec{x} \in \mathbb{R}^m\}\).

Definition (Left Nullspace). The left nullspace of \(A\) is \(\operatorname{Null}(A^T) = \{\vec{x} \in \mathbb{R}^m \mid A^T \vec{x} = \vec{0}\}\).

Definition (Addition and Scalar Multiplication of Mappings). Let \(L, M : \mathbb{R}^n \to \mathbb{R}^m\) be linear and \(c \in \mathbb{R}\). Define \((L + M)(\vec{x}) = L(\vec{x}) + M(\vec{x})\) and \((cL)(\vec{x}) = cL(\vec{x})\).

Definition (Composition). Let \(L : \mathbb{R}^n \to \mathbb{R}^m\) and \(M : \mathbb{R}^m \to \mathbb{R}^p\) be linear. Then \((M \circ L)(\vec{x}) = M(L(\vec{x}))\).

Definition (Vector Space). A set \(V\) with operations of addition \(\vec{x} + \vec{y}\) and scalar multiplication \(c\vec{x}\) is a vector space over \(\mathbb{R}\) if for any \(\vec{v}, \vec{x}, \vec{y} \in V\) and \(a, b \in \mathbb{R}\):
V1: \(\vec{x} + \vec{y} \in V\); V2: associativity of addition; V3: commutativity of addition; V4: existence of zero vector \(\vec{0}\); V5: existence of additive inverse \(-\vec{x}\); V6: \(a\vec{x} \in V\); V7: \(a(b\vec{x}) = (ab)\vec{x}\); V8: \((a+b)\vec{x} = a\vec{x} + b\vec{x}\); V9: \(a(\vec{x}+\vec{y}) = a\vec{x} + a\vec{y}\); V10: \(1\vec{x} = \vec{x}\).

Definition (Subspace). Let \(V\) be a vector space. If \(S \subseteq V\) and \(S\) is a vector space under the same operations as \(V\), then \(S\) is a subspace of \(V\).

Definition (Span in a Vector Space). Let \(B = \{\vec{v}_1, \ldots, \vec{v}_k\}\) be a set of vectors in a vector space \(V\). Then \(\operatorname{Span} B = \{c_1\vec{v}_1 + \cdots + c_k\vec{v}_k \mid c_1, \ldots, c_k \in \mathbb{R}\}\).

Definition (Linear Dependence/Independence in a Vector Space). A set \(\{\vec{v}_1, \ldots, \vec{v}_k\}\) in a vector space \(V\) is linearly dependent if there exist \(c_1, \ldots, c_k\), not all zero, with \(\vec{0} = c_1\vec{v}_1 + \cdots + c_k\vec{v}_k\). It is linearly independent if the only solution is the trivial one.

Definition (Basis). Let \(V\) be a vector space. The set \(B\) is a basis for \(V\) if \(B\) is a linearly independent spanning set for \(V\).

Definition (Dimension). If \(\{\vec{v}_1, \ldots, \vec{v}_n\}\) is a basis for \(V\), then the dimension of \(V\) is \(\dim V = n\). The trivial vector space has dimension 0 (its basis is the empty set). A vector space that has no finite basis is infinite-dimensional.

Definition (Coordinate Vector). Let \(V\) be a vector space with basis \(B = \{\vec{v}_1, \ldots, \vec{v}_n\}\). For \(\vec{v} = b_1\vec{v}_1 + \cdots + b_n\vec{v}_n\), the coordinate vector of \(\vec{v}\) with respect to \(B\) is \([\vec{v}]_B = \begin{pmatrix} b_1 \\ \vdots \\ b_n \end{pmatrix}\).

Definition (Change of Coordinates Matrix). Let \(B = \{\vec{v}_1, \ldots, \vec{v}_n\}\) and \(C\) both be bases for \(V\). The change of coordinates matrix from \(B\)-coordinates to \(C\)-coordinates is \({}_C P_B = [[\vec{v}_1]_C \cdots [\vec{v}_n]_C]\), and \([\vec{x}]_C = {}_C P_B [\vec{x}]_B\).

Definition (Left and Right Inverse). Let \(A\) be \(m \times n\). If \(B\) is \(n \times m\) with \(AB = I_m\), then \(B\) is a right inverse of \(A\). If \(CA = I_n\), then \(C\) is a left inverse of \(A\).

Definition (Inverse / Invertible). Let \(A\) be \(n \times n\). If \(B\) satisfies \(AB = I = BA\), then \(B = A^{-1}\) is the inverse of \(A\), and \(A\) is invertible.

Definition (Elementary Matrix). An \(n \times n\) matrix \(E\) is an elementary matrix if it can be obtained from \(I\) by performing exactly one elementary row operation.

Definition (\(2 \times 2\) Determinant). For \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), \(\det A = ad - bc\).

Definition (Cofactor). Let \(A\) be \(n \times n\) with \(n \geq 2\). Let \(A(i,j)\) be the \((n-1) \times (n-1)\) matrix obtained by deleting the \(i\)-th row and \(j\)-th column. The cofactor of \(a_{ij}\) is \(C_{ij} = (-1)^{i+j} \det A(i,j)\).

Definition (\(n \times n\) Determinant). For an \(n \times n\) matrix \(A\) with \(n \geq 2\), \(\det A = \sum_{i=1}^n a_{i1} C_{i1}\), where \(\det[c] = c\) for \(1 \times 1\) matrices.

Definition (Upper/Lower Triangular). An \(m \times n\) matrix \(U\) is upper triangular if \(u_{ij} = 0\) for \(i > j\). A matrix \(L\) is lower triangular if \(l_{ij} = 0\) for \(i < j\).

Definition (Cofactor Matrix). The cofactor matrix \(\operatorname{cof} A\) has \((\operatorname{cof} A)_{ij} = C_{ij}\).

Definition (Adjugate). The adjugate of \(A\) is \(\operatorname{adj} A = (\operatorname{cof} A)^T\), and \(A^{-1} = \frac{1}{\det A} \operatorname{adj} A\).

Definition (B-Matrix). Let \(B = \{\vec{v}_1, \ldots, \vec{v}_n\}\) be a basis for \(\mathbb{R}^n\) and \(L : \mathbb{R}^n \to \mathbb{R}^n\) a linear operator. The \(B\)-matrix of \(L\) is \([L]_B = [[L(\vec{v}_1)]_B \cdots [L(\vec{v}_n)]_B]\), satisfying \([L(\vec{x})]_B = [L]_B [\vec{x}]_B\).

Definition (Diagonal Matrix). An \(n \times n\) matrix \(D\) is diagonal if \(d_{ij} = 0\) for \(i \neq j\). We write \(D = \operatorname{diag}(d_{11}, \ldots, d_{nn})\).

Definition (Similar Matrices). Matrices \(A\) and \(B\) are similar if there exists an invertible \(P\) with \(P^{-1}AP = B\).

Definition (Eigenvalue, Eigenvector, Eigenpair). Let \(A\) be \(n \times n\). If \(A\vec{v} = \lambda\vec{v}\) for some \(\vec{v} \neq \vec{0}\), then \(\lambda\) is an eigenvalue, \(\vec{v}\) is an eigenvector, and \((\lambda, \vec{v})\) is an eigenpair.

Definition (Eigenvalues/Eigenvectors of a Linear Operator). Let \(L : \mathbb{R}^n \to \mathbb{R}^n\) be a linear operator. If \(L(\vec{v}) = \lambda\vec{v}\) for \(\vec{v} \neq \vec{0}\), then \(\lambda\) is an eigenvalue and \(\vec{v}\) an eigenvector of \(L\).

Definition (Characteristic Polynomial). The characteristic polynomial of an \(n \times n\) matrix \(A\) is \(C(\lambda) = \det(A - \lambda I)\).

Definition (Eigenspace). The eigenspace of the eigenvalue \(\lambda\) is \(E_\lambda = \operatorname{Null}(A - \lambda I)\).

Definition (Algebraic and Geometric Multiplicity). If \(C(\lambda) = (\lambda - \lambda_1)^k C_1(\lambda)\) with \(C_1(\lambda_1) \neq 0\), then the algebraic multiplicity is \(a_{\lambda_1} = k\). The geometric multiplicity is \(g_{\lambda_1} = \dim(E_{\lambda_1})\).

Definition (Diagonalizable). An \(n \times n\) matrix \(A\) is diagonalizable if \(A\) is similar to a diagonal matrix \(D\). If \(P^{-1}AP = D\), then \(P\) diagonalizes \(A\).