Mathematics Linear Algebra Syllabus

What is the multiplicative property of determinants?

$\det(AB)=\det(A)\det(B)$. Also $\det(A^{T})=\det(A)$ and $\det(A^{-1})=\dfrac{1}{\det A}$.

How does the determinant relate to a triangular matrix?

The determinant of a triangular (upper or lower) matrix equals the product of its diagonal entries: $\det = \prod_{i} a_{ii}$.

How does scaling a row by $k$ affect the determinant?

Multiplying one row by a scalar $k$ multiplies the determinant by $k$. Consequently $\det(kA)=k^{n}\det(A)$ for an $n\times n$ matrix.

How does swapping two rows affect the determinant?

It changes the sign of the determinant (multiplies it by $-1$).

Define an eigenvalue and eigenvector of a square matrix $A$.

A nonzero vector $\vec{v}$ is an eigenvector with eigenvalue $\lambda$ if $A\vec{v}=\lambda\vec{v}$. The vector's direction is preserved; only its scale changes.

What is the characteristic equation used to find eigenvalues?

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

What is the relationship between the trace of $A$ and its eigenvalues?

The trace (sum of diagonal entries) equals the sum of the eigenvalues: $\operatorname{tr}(A)=\sum_i \lambda_i$.

What is the relationship between $\det(A)$ and its eigenvalues?

The determinant equals the product of the eigenvalues: $\det(A)=\prod_i \lambda_i$.

What does it mean to diagonalize a matrix $A$?

To write $A=PDP^{-1}$, where $D$ is diagonal containing the eigenvalues and $P$ has the corresponding eigenvectors as columns. Possible when $A$ has $n$ linearly independent eigenvectors.

How can diagonalization simplify computing $A^{k}$?

If $A=PDP^{-1}$, then $A^{k}=PD^{k}P^{-1}$, where $D^{k}$ is found by raising each diagonal eigenvalue to the $k$-th power.

Define the rank of a matrix.

The dimension of its column space (equal to the dimension of its row space) — the number of linearly independent rows or columns, equivalently the number of pivots in row echelon form.

State the Rank–Nullity Theorem.

For an $m\times n$ matrix $A$: $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where nullity is the dimension of the null space and $n$ is the number of columns.

What is the null space (kernel) of a matrix $A$?

The set of all vectors $\vec{x}$ satisfying $A\vec{x}=\vec{0}$. It is a subspace; its dimension is the nullity.

What is the column space of a matrix $A$?

The span of the columns of $A$; equivalently the set of all $\vec{b}$ for which $A\vec{x}=\vec{b}$ has a solution. Its dimension is the rank.

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

A map satisfying $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$ and $T(c\vec{v})=cT(\vec{v})$ — it preserves addition and scalar multiplication.

How is a linear transformation represented as a matrix?

By a matrix $A$ whose $j$-th column is $T(\vec{e}_j)$, the image of the $j$-th standard basis vector. Then $T(\vec{x})=A\vec{x}$.

What is the cross product $\vec{u}\times\vec{v}$ in $\mathbb{R}^{3}$ and its key property?

$\vec{u}\times\vec{v}=\begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix}$, a vector orthogonal to both $\vec{u}$ and $\vec{v}$ with magnitude $\|\vec{u}\|\|\vec{v}\|\sin\theta$.

What does Cramer's Rule give for the solution of $A\vec{x}=\vec{b}$?

$x_i=\dfrac{\det(A_i)}{\det(A)}$, where $A_i$ is $A$ with its $i$-th column replaced by $\vec{b}$. Valid when $\det A \neq 0$.

What is the goal of Gaussian elimination?

To reduce a matrix to row echelon form using elementary row operations (row swaps, scaling, adding multiples of rows), to solve linear systems, find rank, or compute inverses.

What are the three elementary row operations?

1) Swap two rows. 2) Multiply a row by a nonzero scalar. 3) Add a scalar multiple of one row to another row.

Define an orthogonal matrix and give its key property.

A square matrix $Q$ with $Q^{T}Q=I$, so $Q^{-1}=Q^{T}$. Its columns are orthonormal, it preserves lengths and angles, and $\det Q = \pm 1$.

What characterizes an orthonormal set of vectors?

Each vector has unit length and every pair is orthogonal: $\vec{u}_i\cdot\vec{u}_j = \delta_{ij}$ (1 if $i=j$, 0 otherwise).

What is the projection of $\vec{u}$ onto $\vec{v}$?

$\operatorname{proj}_{\vec{v}}\vec{u}=\dfrac{\vec{u}\cdot\vec{v}}{\vec{v}\cdot\vec{v}}\,\vec{v}=\dfrac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^{2}}\,\vec{v}$.

State the Cauchy–Schwarz inequality.

$|\vec{u}\cdot\vec{v}|\leq \|\vec{u}\|\,\|\vec{v}\|$, with equality if and only if $\vec{u}$ and $\vec{v}$ are linearly dependent (parallel).

State the triangle inequality for vector norms.

$\|\vec{u}+\vec{v}\|\leq \|\vec{u}\|+\|\vec{v}\|$.

What does the Gram–Schmidt process do?

It converts a linearly independent set $\{\vec{v}_1,\dots,\vec{v}_n\}$ into an orthogonal (or orthonormal) set spanning the same subspace by successively subtracting projections onto previously found vectors.

What is the trace of a square matrix and a key property?

$\operatorname{tr}(A)=\sum_i A_{ii}$, the sum of diagonal entries. Key property: $\operatorname{tr}(AB)=\operatorname{tr}(BA)$, and trace is invariant under similarity.

When does the homogeneous system $A\vec{x}=\vec{0}$ have nontrivial solutions?

When $\det A = 0$ (for square $A$), equivalently when the columns are linearly dependent or $\operatorname{rank}(A)<n$ (number of unknowns).

State the Cayley–Hamilton theorem.

Every square matrix satisfies its own characteristic equation: if $p(\lambda)=\det(A-\lambda I)$, then $p(A)=0$ (the zero matrix).

What is the spectral theorem for real symmetric matrices?

Every real symmetric matrix $A$ is orthogonally diagonalizable: $A=QDQ^{T}$ with $Q$ orthogonal and $D$ diagonal of real eigenvalues. Its eigenvectors can be chosen orthonormal.

Compare the singular value decomposition (SVD) to eigen-decomposition.

SVD factors any $m\times n$ matrix as $A=U\Sigma V^{T}$ with $U,V$ orthogonal and $\Sigma$ diagonal of nonnegative singular values; it works for non-square/non-diagonalizable matrices, whereas eigen-decomposition $A=PDP^{-1}$ requires a square, diagonalizable matrix.