Linear Algebra LaTeX

Some basic LaTeX snippets common used in Linear Algebra

1. Vector Notation

$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$

\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}

2. Matrix Notation

$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

A = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}

3. Dot Product

$\vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i$

\vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i

4. Cross Product (3D vectors)

$\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

\vec{u} \times \vec{v} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}

5. Determinant of a 2x2 Matrix

$\det\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = ad - bc$

\det\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = ad - bc

6. Identity Matrix

$I = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}$

I = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}

7. Inverse of a Matrix (if it exists)

$A^{-1} \quad \text{such that} \quad AA^{-1} = I$

A^{-1} \quad \text{such that} \quad AA^{-1} = I

8. Eigenvalue Equation

$A\vec{v} = \lambda \vec{v}$

A\vec{v} = \lambda \vec{v}

9. Rank of a Matrix

$\text{rank}(A)$

\text{rank}(A)