Linear Algebra

What is the rectangular form of a complex number $z$?

$z = a + bi$, where $a = \text{Re}(z)$ is the real part and $b = \text{Im}(z)$ is the imaginary part, with $i^2 = -1$.

What is the modulus of a complex number $z = a + bi$?

The modulus is $\lvert z \rvert = \sqrt{a^2 + b^2}$, representing the distance from the origin to $z$ in the complex plane.

What is the principal argument $\text{Arg}(z)$ of a complex number $z \neq 0$?

The unique angle $\theta$ in the interval $(-\pi, \pi]$ measured counterclockwise from the positive real axis to the line segment connecting the origin to $z$. It is undefined for $z = 0$.

What is Euler's formula for a complex number?

$e^{i\theta} = \cos\theta + i\sin\theta$, allowing any complex number to be written in polar form $z = r e^{i\theta}$, where $r = \lvert z \rvert$ and $\theta = \arg(z)$.

What is De Moivre's Theorem for integer powers of a complex number?

For any integer $n$ and $z = r(\cos\theta + i\sin\theta)$, $z^n = r^n (\cos(n\theta) + i\sin(n\theta)) = r^n e^{in\theta}$.

What is the formula for the $n$-th roots of a complex number $w = R e^{i\phi}$?

The $n$ distinct roots are $z_k = \sqrt[n]{R} \exp\left(i \frac{\phi + 2k\pi}{n}\right)$ for $k = 0, 1, 2, \dots, n-1$.

What is a system of $m$ linear equations in $n$ unknowns in matrix form?

$A\mathbf{x} = \mathbf{b}$, where $A$ is an $m \times n$ coefficient matrix, $\mathbf{x}$ is an $n \times 1$ vector of unknowns, and $\mathbf{b}$ is an $m \times 1$ right-hand side vector.

What are the three elementary row operations (EROs) on a matrix?

1. Swap two rows ($R_i \leftrightarrow R_j$). 2. Multiply a row by a non-zero scalar ($c R_i \to R_i$, $c \neq 0$). 3. Add a scalar multiple of one row to another ($R_j + c R_i \to R_j$).

What conditions define a matrix in Row Echelon Form (REF)?

1. All zero rows are at the bottom. 2. The leading entry (pivot) of each non-zero row is strictly to the right of the leading entry of the row above it. 3. All entries below a leading entry are zero.

What additional conditions define Reduced Row Echelon Form (RREF)?

In addition to the REF conditions: 1. Every leading entry is equal to $1$. 2. Each column containing a leading $1$ has zeros in all other positions.

What is the System Consistency Theorem for $A\mathbf{x} = \mathbf{b}$?

A linear system $A\mathbf{x} = \mathbf{b}$ is consistent if and only if $\mathbf{b}$ can be expressed as a linear combination of the column vectors of $A$ (that is, $\mathbf{b} \in \text{col}(A)$).

Under what condition does a linear system $A\mathbf{x} = \mathbf{b}$ have infinitely many solutions?

When the system is consistent and has at least one free variable (the number of unknowns $n$ is strictly greater than the number of pivot columns).

What is the parametric vector form of the general solution to $A\mathbf{x} = \mathbf{b}$?

$\mathbf{x} = \mathbf{p} + t_1 \mathbf{v}_1 + t_2 \mathbf{v}_2 + \dots + t_k \mathbf{v}_k$, where $\mathbf{p}$ is a particular solution and $\mathbf{v}_1, \dots, \mathbf{v}_k$ form a basis for the solution space of the homogeneous system $A\mathbf{x} = \mathbf{0}$.

When is the product $AB$ of two matrices defined?

When the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.

What is the row-column rule for the entry $(AB)_{ij}$ of matrix product $AB$?

$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj}$ (row $i$ of $A$ times column $j$ of $B$).

Is matrix multiplication commutative in general?

No, in general $AB \neq BA$, so the order of the factors cannot be interchanged.

Does the cancellation law hold for matrix multiplication?

No, $AB = AC$ does not imply $B = C$ in general, even if $A \neq \mathbf{0}$. It does hold if $A$ is invertible (multiply both sides by $A^{-1}$).

Does $AB = \mathbf{0}$ imply $A = \mathbf{0}$ or $B = \mathbf{0}$?

No, two non-zero matrices can multiply to give the zero matrix.

What is the transpose $A^T$ of an $m \times n$ matrix $A$?

The $n \times m$ matrix obtained by interchanging the rows and columns of $A$, so that $(A^T)_{ij} = a_{ji}$.

What is the formula for the transpose of a matrix product $(AB)^T$?

$(AB)^T = B^T A^T$, so the order of multiplication is reversed.

What is the trace $\text{tr}(A)$ of a square $n \times n$ matrix $A$?

The sum of the main diagonal entries: $\text{tr}(A) = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}$.

What property connects the trace of matrix products $\text{tr}(AB)$ and $\text{tr}(BA)$?

$\text{tr}(AB) = \text{tr}(BA)$ whenever both products are defined (if $A$ is $m \times n$ and $B$ is $n \times m$, both products are square).

What is the definition of an invertible (non-singular) matrix $A$?

An $n \times n$ matrix $A$ is invertible if there exists an $n \times n$ matrix $B$ such that $AB = BA = I_n$. Then $B = A^{-1}$.

What is the inverse formula for a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, defined if and only if $\det(A) = ad - bc \neq 0$.

What is the formula for the inverse of a matrix product $(AB)^{-1}$?

$(AB)^{-1} = B^{-1} A^{-1}$, assuming $A$ and $B$ are invertible matrices of the same size.

What is an elementary matrix $E$?

A square matrix obtained from the identity matrix $I_n$ by performing a single elementary row operation (ERO).

How does left-multiplication by an elementary matrix $E$ affect a matrix $A$?

$EA$ equals the matrix that results from performing on $A$ the same ERO that produced $E$ from $I_n$.

Are elementary matrices invertible, and what is the form of $E^{-1}$?

Every elementary matrix $E$ is invertible, and $E^{-1}$ is the elementary matrix of the reverse ERO (swap back, scale by $1/c$, or subtract the added multiple).

What is the Fundamental Theorem on Elementary Matrices and Invertibility?

A square matrix $A$ is invertible if and only if it is a product of elementary matrices. If $E_k \cdots E_2 E_1 A = I$, then $A = E_1^{-1} E_2^{-1} \cdots E_k^{-1}$ and $A^{-1} = E_k \cdots E_2 E_1$.

What are the equivalent statements of the Invertible Matrix Theorem for an $n \times n$ matrix $A$?

1. $A$ is invertible. 2. $A\mathbf{x} = \mathbf{0}$ has only the trivial solution $\mathbf{x} = \mathbf{0}$. 3. The RREF of $A$ is $I_n$. 4. $A$ is a product of elementary matrices. 5. $A\mathbf{x} = \mathbf{b}$ is consistent for every $\mathbf{b} \in \mathbb{R}^n$. 6. $A\mathbf{x} = \mathbf{b}$ has the unique solution $\mathbf{x} = A^{-1}\mathbf{b}$. 7. $\det(A) \neq 0$.

When is a diagonal matrix $D = \text{diag}(d_1, d_2, \dots, d_n)$ invertible?

When all its diagonal entries are non-zero ($d_i \neq 0$ for all $i$), in which case $D^{-1} = \text{diag}(1/d_1, 1/d_2, \dots, 1/d_n)$.

What is a symmetric matrix $A$?

A square matrix $A$ that is equal to its transpose: $A^T = A$ (that is, $a_{ij} = a_{ji}$ for all $i, j$).

What is an $LU$-decomposition of a matrix $A$?

A factorization $A = LU$, where $L$ is a unit lower triangular matrix (diagonal entries equal to $1$) and $U$ is an upper triangular matrix in row echelon form. It exists when $A$ can be reduced to REF without row swaps.

How is $A\mathbf{x} = \mathbf{b}$ solved using $A = LU$?

1. Solve $L\mathbf{y} = \mathbf{b}$ for $\mathbf{y}$ by forward substitution. 2. Solve $U\mathbf{x} = \mathbf{y}$ for $\mathbf{x}$ by back substitution.

What is the minor $M_{ij}$ of an entry $a_{ij}$ in a square matrix $A$?

The determinant of the $(n-1) \times (n-1)$ submatrix that remains after deleting row $i$ and column $j$ from $A$.

What is the cofactor $C_{ij}$ of an entry $a_{ij}$ in a square matrix $A$?

$C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor of $a_{ij}$. The sign is $+$ when $i + j$ is even and $-$ when it is odd.

What is Laplace's Cofactor Expansion Theorem for $\det(A)$?

The determinant of an $n \times n$ matrix equals the sum of the products of the entries of any single row (or column) with their cofactors: $\det(A) = \sum_{j=1}^{n} a_{ij} C_{ij}$ (row $i$) $= \sum_{i=1}^{n} a_{ij} C_{ij}$ (column $j$).

What is the determinant of a triangular or diagonal matrix?

The product of the entries on its main diagonal: $\det(A) = a_{11} a_{22} \cdots a_{nn}$.

How does swapping two rows of a matrix affect its determinant?

Interchanging two rows negates the determinant: $\det(B) = -\det(A)$.

How does multiplying a single row of a matrix by scalar $c$ affect its determinant?

Scaling one row by $c$ scales the determinant by $c$: $\det(B) = c \det(A)$.

How does adding a scalar multiple of one row to another row affect the determinant?

It leaves the determinant unchanged: $\det(B) = \det(A)$.

What is the formula for the determinant of a scalar multiple $cA$ of an $n \times n$ matrix $A$?

$\det(cA) = c^n \det(A)$, where $n$ is the size of the matrix.

What is the determinant product rule for matrices $A$ and $B$?

$\det(AB) = \det(A) \det(B)$ for square matrices of the same size.

What is the relation between $\det(A^T)$ and $\det(A)$?

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

What is the determinant of an inverse matrix $A^{-1}$?

$\det(A^{-1}) = \frac{1}{\det(A)}$, provided $\det(A) \neq 0$.

What is the classical adjoint $\text{adj}(A)$ of a square matrix $A$?

The transpose of the matrix of cofactors $C$ of $A$: $\text{adj}(A) = C^T$, where $(C)_{ij} = C_{ij}$.

What formula expresses $A^{-1}$ using the classical adjoint $\text{adj}(A)$?

$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, valid for any invertible matrix $A$.

What is Cramer's Rule for solving $A\mathbf{x} = \mathbf{b}$?

If $\det(A) \neq 0$, the unique solution has entries $x_j = \frac{\det(A_j)}{\det(A)}$, where $A_j$ is the matrix obtained by replacing column $j$ of $A$ with $\mathbf{b}$.

What is the Cauchy-Schwarz Inequality in Euclidean $n$-space $\mathbb{R}^n$?

$\lvert \mathbf{u} \cdot \mathbf{v} \rvert \le \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert$ for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$.

What is the Triangle Inequality for vectors in $\mathbb{R}^n$?

$\lVert \mathbf{u} + \mathbf{v} \rVert \le \lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$ for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$.

What is the geometric definition of the cross product $\mathbf{u} \times \mathbf{v}$ in $\mathbb{R}^3$?

A vector orthogonal to both $\mathbf{u}$ and $\mathbf{v}$ with magnitude $\lVert \mathbf{u} \times \mathbf{v} \rVert = \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert \sin\theta$, directed by the right-hand rule.

What is the area of the parallelogram determined by vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^3$?

$\text{Area} = \lVert \mathbf{u} \times \mathbf{v} \rVert$, the norm of their cross product.

What is the triple scalar product and its geometric meaning?

$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$. Its absolute value $\lvert \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \rvert$ equals the volume of the parallelepiped determined by $\mathbf{u}, \mathbf{v}, \mathbf{w}$.

What are the 10 axioms of a real vector space $V$?

1. Closed under addition. 2. $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$. 3. $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$. 4. There is a zero vector with $\mathbf{u}+\mathbf{0}=\mathbf{u}$. 5. Each $\mathbf{u}$ has an additive inverse with $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$. 6. Closed under scalar multiplication. 7. $k(\mathbf{u}+\mathbf{v})=k\mathbf{u}+k\mathbf{v}$. 8. $(k+m)\mathbf{u}=k\mathbf{u}+m\mathbf{u}$. 9. $k(m\mathbf{u})=(km)\mathbf{u}$. 10. $1\mathbf{u}=\mathbf{u}$.

What is the Subspace Test Theorem for a subset $W$ of a vector space $V$?

A non-empty subset $W$ is a subspace of $V$ if and only if: 1. $W$ is closed under addition ($\mathbf{u}, \mathbf{v} \in W \implies \mathbf{u}+\mathbf{v} \in W$). 2. $W$ is closed under scalar multiplication ($\mathbf{u} \in W, k \in \mathbb{R} \implies k\mathbf{u} \in W$).

Is the solution set of a homogeneous linear system $A\mathbf{x} = \mathbf{0}$ a subspace?

Yes, the solution set of $A\mathbf{x} = \mathbf{0}$ ($A$ is $m \times n$) is always a subspace of $\mathbb{R}^n$, called the null space $\text{null}(A)$.

Is the solution set of a non-homogeneous linear system $A\mathbf{x} = \mathbf{b}$ ($\mathbf{b} \neq \mathbf{0}$) a subspace?

No, because it does not contain the zero vector $\mathbf{0}$ and is not closed under addition or scalar multiplication.

What is the span $\text{span}(S)$ of a set of vectors $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$?

The set of all linear combinations of vectors in $S$: $\text{span}(S) = \{c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k : c_i \in \mathbb{R}\}$. It is the smallest subspace containing $S$.

What is the definition of linear independence for a set of vectors $S = \{\mathbf{v}_1, \dots, \mathbf{v}_k\}$?

$S$ is linearly independent if the vector equation $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ has ONLY the trivial solution $c_1 = c_2 = \dots = c_k = 0$.

What defines a linearly dependent set of vectors?

A set for which $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ has a solution with at least one $c_i \neq 0$. Equivalently, at least one vector is a linear combination of the others.

Is any set containing the zero vector $\mathbf{0}$ linearly independent?

No, any finite set containing the zero vector $\mathbf{0}$ is linearly dependent.

What is a basis for a vector space $V$?

A set of vectors $B = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ in $V$ that is both linearly independent and spans $V$ ($\text{span}(B) = V$).

What is the Unique Representation Theorem for a basis $B$?

If $B = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ is a basis for $V$, then every vector $\mathbf{v} \in V$ can be expressed in exactly one way as a linear combination of the vectors in $B$.

What is the dimension $\dim(V)$ of a finite-dimensional vector space $V$?

The number of vectors in any basis for $V$ (all bases have the same number). The dimension of the zero vector space $\{\mathbf{0}\}$ is defined to be $0$.

What is the Plus/Minus Theorem for linear independence and span?

1. If $S$ is linearly independent and $\mathbf{v} \notin \text{span}(S)$, then $S \cup \{\mathbf{v}\}$ is linearly independent. 2. If $\mathbf{v} \in S$ is a linear combination of the other vectors in $S$, then $\text{span}(S \setminus \{\mathbf{v}\}) = \text{span}(S)$.

What are the row space $\text{row}(A)$ and column space $\text{col}(A)$ of an $m \times n$ matrix $A$?

$\text{row}(A)$ is the subspace of $\mathbb{R}^n$ spanned by the row vectors of $A$. $\text{col}(A)$ is the subspace of $\mathbb{R}^m$ spanned by the column vectors of $A$.

What is the null space $\text{null}(A)$ of an $m \times n$ matrix $A$?

The subspace of $\mathbb{R}^n$ consisting of all solutions to the homogeneous linear system $A\mathbf{x} = \mathbf{0}$.

What is the Rank-Nullity Theorem for an $m \times n$ matrix $A$?

$\text{rank}(A) + \text{nullity}(A) = n$, where $n$ is the number of columns of $A$, $\text{rank}(A) = \dim(\text{col}(A))$ and $\text{nullity}(A) = \dim(\text{null}(A))$.

$$\begin{gathered} \begin{bmatrix} \color{#60a5fa}1 & \color{#fb923c}2 & \color{#60a5fa}0 \\ \color{#60a5fa}0 & \color{#fb923c}0 & \color{#60a5fa}1 \end{bmatrix} \\[10pt] \underbrace{\color{#60a5fa}2}_{\displaystyle\color{#60a5fa}\text{rank}} \;+\; \underbrace{\color{#fb923c}1}_{\displaystyle\color{#fb923c}\text{nullity}} \;=\; \underbrace{3}_{\displaystyle\text{columns}} \end{gathered}$$

How do elementary row operations affect the row space, column space, and null space of a matrix $A$?

EROs preserve the row space ($\text{row}(A) = \text{row}(R)$) and the null space ($\text{null}(A) = \text{null}(R)$), but they can change the column space.

How do you construct a basis for $\text{col}(A)$?

Identify the pivot columns in the REF/RREF of $A$, then take the corresponding columns of the ORIGINAL matrix $A$.

How do you construct a basis for $\text{row}(A)$?

Take the non-zero row vectors of any Row Echelon Form (REF) or RREF of $A$.

What is a transition (change-of-basis) matrix $P_{B \to B'}$?

The matrix whose columns are the coordinate vectors of the old basis vectors relative to the new basis: $P_{B \to B'} = \begin{bmatrix} [\mathbf{b}_1]_{B'} & [\mathbf{b}_2]_{B'} & \cdots & [\mathbf{b}_n]_{B'} \end{bmatrix}$.

What is the relationship between coordinate vectors under a change of basis $P_{B \to B'}$?

$[\mathbf{v}]_{B'} = P_{B \to B'} [\mathbf{v}]_B$, and $P_{B \to B'}$ is invertible with $(P_{B \to B'})^{-1} = P_{B' \to B}$.

What are the 4 axioms of a real inner product $\langle \mathbf{u}, \mathbf{v} \rangle$ on a vector space $V$?

1. Symmetry: $\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle$. 2. Additivity: $\langle \mathbf{u} + \mathbf{v}, \mathbf{w} \rangle = \langle \mathbf{u}, \mathbf{w} \rangle + \langle \mathbf{v}, \mathbf{w} \rangle$. 3. Homogeneity: $\langle k\mathbf{u}, \mathbf{v} \rangle = k\langle \mathbf{u}, \mathbf{v} \rangle$. 4. Positivity: $\langle \mathbf{v}, \mathbf{v} \rangle \ge 0$, and $\langle \mathbf{v}, \mathbf{v} \rangle = 0 \iff \mathbf{v} = \mathbf{0}$.

What is the norm $\lVert \mathbf{u} \rVert$ induced by a general inner product $\langle \mathbf{u}, \mathbf{v} \rangle$?

$\lVert \mathbf{u} \rVert = \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle}$.

What is the definition of orthogonal vectors in a general inner product space?

Vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$.

What is the Generalized Pythagorean Theorem in an inner product space?

Vectors $\mathbf{u}$ and $\mathbf{v}$ in a real inner product space are orthogonal if and only if $\lVert \mathbf{u} + \mathbf{v} \rVert^2 = \lVert \mathbf{u} \rVert^2 + \lVert \mathbf{v} \rVert^2$.

What is the formula for the orthogonal projection of $\mathbf{u}$ onto a vector $\mathbf{v} \neq \mathbf{0}$?

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\lVert \mathbf{v} \rVert^2} \mathbf{v}$.

What defines an orthonormal basis $B = \{\mathbf{q}_1, \mathbf{q}_2, \dots, \mathbf{q}_n\}$?

A basis whose vectors are mutually orthogonal ($\langle \mathbf{q}_i, \mathbf{q}_j \rangle = 0$ for $i \neq j$) and normalized ($\lVert \mathbf{q}_i \rVert = 1$ for all $i$).

What is the coordinate expansion formula for a vector $\mathbf{u}$ relative to an orthonormal basis $B = \{\mathbf{q}_1, \dots, \mathbf{q}_n\}$?

$\mathbf{u} = \langle \mathbf{u}, \mathbf{q}_1 \rangle \mathbf{q}_1 + \langle \mathbf{u}, \mathbf{q}_2 \rangle \mathbf{q}_2 + \dots + \langle \mathbf{u}, \mathbf{q}_n \rangle \mathbf{q}_n$.

What are the Gram-Schmidt steps to convert a basis $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$ into an orthogonal basis $\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$?

$\mathbf{v}_1 = \mathbf{u}_1$, then $\mathbf{v}_2 = \mathbf{u}_2 - \frac{\langle \mathbf{u}_2, \mathbf{v}_1 \rangle}{\lVert \mathbf{v}_1 \rVert^2}\mathbf{v}_1$, and in general $\mathbf{v}_k = \mathbf{u}_k - \sum_{j=1}^{k-1} \frac{\langle \mathbf{u}_k, \mathbf{v}_j \rangle}{\lVert \mathbf{v}_j \rVert^2}\mathbf{v}_j$.

What is a $QR$-decomposition of an $m \times n$ matrix $A$ with linearly independent columns?

A factorization $A = QR$, where $Q$ is an $m \times n$ matrix with orthonormal columns ($Q^T Q = I_n$) obtained by Gram-Schmidt, and $R = Q^T A$ is an $n \times n$ invertible upper triangular matrix.

What is the Best Approximation Theorem for a subspace $W$?

For any vector $\mathbf{u} \in V$, the orthogonal projection $\text{proj}_W \mathbf{u}$ is the unique closest vector in $W$ to $\mathbf{u}$: $\lVert \mathbf{u} - \text{proj}_W \mathbf{u} \rVert < \lVert \mathbf{u} - \mathbf{w} \rVert$ for all $\mathbf{w} \in W$ with $\mathbf{w} \neq \text{proj}_W \mathbf{u}$.

What are the Normal Equations for the least squares problem $A\mathbf{x} = \mathbf{b}$?

$A^T A \mathbf{x}^* = A^T \mathbf{b}$. If $A$ has linearly independent columns, the unique least squares solution is $\mathbf{x}^* = (A^T A)^{-1} A^T \mathbf{b}$.

What defines a linear transformation $T: V \to W$?

A mapping such that for all $\mathbf{u}, \mathbf{v} \in V$ and every scalar $c$: 1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$. 2. $T(c\mathbf{u}) = c T(\mathbf{u})$.

What are the basic properties of every linear transformation $T: V \to W$?

1. $T(\mathbf{0}_V) = \mathbf{0}_W$. 2. $T(-\mathbf{u}) = -T(\mathbf{u})$ and $T(\mathbf{u} - \mathbf{v}) = T(\mathbf{u}) - T(\mathbf{v})$. 3. $T(c_1\mathbf{u}_1 + \dots + c_k\mathbf{u}_k) = c_1 T(\mathbf{u}_1) + \dots + c_k T(\mathbf{u}_k)$.

What is the kernel (null space) $\ker(T)$ of a linear transformation $T: V \to W$?

The subspace of $V$ consisting of all vectors mapped to the zero vector of $W$: $\ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}_W\}$.

What is the range (image) $\text{range}(T)$ of a linear transformation $T: V \to W$?

The subspace of $W$ consisting of all images of vectors in $V$: $\text{range}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\}$.

What is the Rank-Nullity Theorem for a linear transformation $T: V \to W$ on a finite-dimensional space $V$?

$\text{nullity}(T) + \text{rank}(T) = \dim(V)$, where $\text{nullity}(T) = \dim(\ker(T))$ and $\text{rank}(T) = \dim(\text{range}(T))$.

When is a linear transformation $T: V \to W$ one-to-one (injective)?

$T$ is one-to-one if $T(\mathbf{u}) = T(\mathbf{v}) \implies \mathbf{u} = \mathbf{v}$. Equivalently, $T$ is one-to-one if and only if $\ker(T) = \{\mathbf{0}_V\}$ (that is, $\text{nullity}(T) = 0$).

When is a linear transformation $T: V \to W$ onto (surjective)?

$T$ is onto if $\text{range}(T) = W$. For finite-dimensional $W$, this means $\text{rank}(T) = \dim(W)$.

What is the standard matrix $[T]$ for a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$?

The $m \times n$ matrix $[T] = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{bmatrix}$, where $\mathbf{e}_i$ are the standard basis vectors of $\mathbb{R}^n$, so that $T(\mathbf{x}) = [T]\mathbf{x}$.

$\text{Example: Let } T(x,y) = (2x+y, x-y) \\</p><p>\text{1. } T(\mathbf{e}_1) = T(1,0) = (2, 1) \\</p><p>\text{2. } T(\mathbf{e}_2) = T(0,1) = (1, -1) \\</p><p>\therefore [T] = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}$

What is an eigenvalue $\lambda$ and eigenvector $\mathbf{x}$ of a square matrix $A$?

A scalar $\lambda$ and a NON-ZERO vector $\mathbf{x} \neq \mathbf{0}$ satisfying $A\mathbf{x} = \lambda \mathbf{x}$.

What is the characteristic equation of an $n \times n$ matrix $A$?

$\det(\lambda I_n - A) = 0$ (equivalently $\det(A - \lambda I_n) = 0$). Its roots are the eigenvalues of $A$.

What is the eigenspace $E_\lambda$ corresponding to an eigenvalue $\lambda$ of $A$?

The solution space of the homogeneous system $(\lambda I_n - A)\mathbf{x} = \mathbf{0}$, consisting of the zero vector and all eigenvectors for $\lambda$: $E_\lambda = \text{null}(\lambda I_n - A)$.

What is the relationship between algebraic multiplicity and geometric multiplicity of an eigenvalue $\lambda$?

The algebraic multiplicity is the number of times $\lambda$ is a root of the characteristic polynomial. The geometric multiplicity is $\dim(E_\lambda)$. For every eigenvalue, $1 \le \text{geometric multiplicity} \le \text{algebraic multiplicity}$.

What does it mean for two $n \times n$ matrices $A$ and $B$ to be similar?

$B = P^{-1} A P$ for some invertible $n \times n$ matrix $P$. Similar matrices share the same determinant, trace, rank, characteristic polynomial and eigenvalues.

What is the Matrix Diagonalization Theorem?

An $n \times n$ matrix $A$ is diagonalizable if and only if $A$ has $n$ linearly independent eigenvectors. Then $A = P D P^{-1}$, where $D = \text{diag}(\lambda_1, \dots, \lambda_n)$ and the columns of $P$ are the corresponding eigenvectors.

How do you compute powers $A^k$ of a diagonalizable matrix $A = P D P^{-1}$?

$A^k = P D^k P^{-1} = P \begin{bmatrix} \lambda_1^k & 0 & \cdots & 0 \\ 0 & \lambda_2^k & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n^k \end{bmatrix} P^{-1}$.

What are the spectral properties of real symmetric matrices ($A^T = A$)?

1. All eigenvalues of a real symmetric matrix are real numbers. 2. Eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.