Linear Algebra Cheat Sheet (AI-GENERATED)

EIGENVALUES AND EIGENVECTORS OF A MATRIX

Key Concepts:

• Eigenvalues ($\lambda$) are scalars such that when multiplied by a vector ($v$), it is equivalent to the linear transformation of $v$ by matrix $A$, i.e., $Av=\lambda v$.
• Eigenvectors ($v$) are non-zero vectors that change by only a scalar factor when that linear transformation is applied.
• Characteristic Equation: $det(A-\lambda I)=0$, where $I$ is the identity matrix of the same dimension as $A$. Solving this gives the eigenvalues.
• For each eigenvalue, find the eigenvector(s) by solving $(A-\lambda I)v=0$.

Diagonalisation:

• A matrix $A$ is diagonalisable if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A=PD{P}^{-1}$.
• Conditions:
  • $A$ must have $n$ linearly independent eigenvectors.
  • If $A$ is $n\times n$, there must be $n$ eigenvalues (counting multiplicities).

SOLVING LINEAR SYSTEMS (HOMOGENEOUS)

Key Concepts:

• A system is homogeneous if it can be written as $Ax=0$.
• Null Space (Null(A)): The set of all vectors $x$ that satisfy $Ax=0$. It’s a vector subspace.
• Basis of Null Space: The set of linearly independent vectors that span the Null Space.

Gauss-Jordan Elimination:

• A method to solve linear systems, find the rank, and determine the inverse of a matrix.
• Perform row operations to transform the matrix into reduced row echelon form (RREF).
• For $Ax=0$, RREF reveals the free variables, which help express the solution set.

INVERSE OF A MATRIX

Key Concepts:

• The inverse of an $n\times n$ matrix $A$ is a matrix $A^{-1}$ such that $A{A}^{-1}=A^{-1}A=I$.
• A matrix is invertible (non-singular) if it has an inverse, which occurs iff |A| (determinant of $A$) $\ne 0$.

Finding the Inverse:

• Method 1: Gauss-Jordan Elimination - Augment $A$ with $I$ and perform row operations until $A$ becomes $I$, and $I$ becomes $A^{-1}$.
• Method 2: Adjugate Formula - $A^{-1}=\frac{1}{det(A)}\cdot adj(A)$ where $adj(A)$ is the adjugate (transpose of the cofactor matrix) of $A$.

BASIS OF A VECTOR SPACE & BASIS PROOF

Key Concepts:

• A basis of a vector space is a set of vectors in the space that is linearly independent and spans the vector space.
• Spanning Set: A set of vectors that can combine to form any vector in the space.
• Linearly Independent: No vector in the set can be written as a combination of the others.

Proving a Basis:

• To prove that $n$ vectors form a basis of ${\mathbb{R}}^n$, show that:
  • They are linearly independent.
  • They span ${\mathbb{R}}^n$, typically by showing the only solution to their homogeneous system is the trivial solution, indicating independence, and counting the vectors matches the dimension of the space for spanning.