Linear Algebra Mid-Term Prediction (Cheat Sheet)

MTH1004 Linear Algebra Cheat Sheet by William Fayers :)

Linear Algebra Exam Cheat Sheet

Eigenvalues and Eigenvectors

• Characteristic Equation: $\det (A-\lambda I)=0$
• Eigenvector: Non-zero vector $v$ such that $Av=\lambda v$, where $\lambda$ is an eigenvalue. To solve, first re-arrange to $(A-\lambda I)v=0$, simplify the left matrix, then solve that equation with Gauss-Jordan Elimination. However, it’s pretty sensible to check the solution in the original formula form, too (especially if you don’t find all the values for a vector, you can assume the missing ones are $1$ and then check it).

Diagonalisation

• Condition: A matrix $A$ is diagonalisable if it has $n$ linearly independent eigenvectors.
• Process:
  1. Find eigenvalues ($\lambda$) using the characteristic equation.
  2. For each $\lambda$, find $(A-\lambda I)v=0$.
  3. Form $P$ from eigenvectors and $D$ (the result) with eigenvalues on the diagonal.
  4. $P^{-1}AP=D$.
• Goal: Form an equivalent matrix with the the eigenvalues on the diagonal.

Solving Linear Systems

• Gauss-Jordan Elimination: Process for putting a matrix into Reduced Row Echelon Form (RREF).
• Homogeneous Systems: $Ax=0$, solution is vector subspace.
• Null Space: Set of all solutions to $Ax=0$; basis found during RREF.

Matrix Inverse and Gauss-Jordan

• Finding $A^{-1}$: Augment $A$ with $I$, apply Gauss-Jordan to transform $A$ into $I$, resulting matrix next to $I$ is $A^{-1}$.
• Condition for Invertibility: $A$ is invertible if and only if $\det (A)\ne 0$.

Basis and Dimension

• Basis of Vector Space: Linearly independent set that spans the space.
• To Prove Basis: Show set is linearly independent and spans vector space.
• Dimension: Number of vectors in a basis for the space.

$\mathbf{u},\mathbf{v}\in {\mathbb{R}}^2$ are linearly independent if, for $c_1,c_2\in \mathbb{R}$:

$$c_1\mathbf{u}+c_2\mathbf{v}=0⟹c_1=c_2=0$$

Cramer’s Rule

Suppose you have a system of $n$ linear equations with $n$ unknowns, represented as:

$$\begin{array}{rl}{a}_{11}{x}_1+a_{12}{x}_2+\dots +a_{1n}{x}_n& =b_1\\ a_{21}{x}_1+a_{22}{x}_2+\dots +a_{2n}{x}_n& =b_2\\ ⋮& \\ a_{n1}{x}_1+a_{n2}{x}_2+\dots +a_{nn}{x}_n& =b_n\end{array}$$

This system can be written in matrix form as $AX=B$, where:

• $A$ is the coefficient matrix,
• $X$ is the column matrix of variables $(x_1,x_2,\dots ,x_n{)}^T$,
• $B$ is the column matrix of constants $(b_1,b_2,\dots ,b_n{)}^T$.

Cramer’s rule states that the solution for each variable $x_i$ is given by:

$$x_i=\frac{\det (A_i)}{\det (A)}$$

where $\det (A)$ is the determinant of the coefficient matrix $A$, and $\det (A_i)$ is the determinant of the matrix formed by replacing the $i$-th column of $A$ with the column matrix $B$.

Additional Tips

• Determinant Properties:
  • $\det (AB)=\det (A)\det (B)$
  • $\det (A^{-1})=1/\det (A)$
  • $\det (P^{-1}AP)=\det (A)$
• Rank Theorem: $\text{Rank}(A)+\text{Nullity}(A)=n$, where $n$ is the number of columns in $A$, $\text{Rank}(A)$ is the number of pivots in row-echelon form, $\text{Nullity}(A)$ is the dimension of $\text{Null}(A)$.
• Multiplicities: An algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic equation of a matrix, whilst a geometric multiplicity is the dimension of the eigenspace corresponding to an eigenvalue.
• Column Space: $\text{Col}(A)$ is the set of all possible linear combinations of its column vectors, given that they’re all independent.
• Multiplying Matrices: For a 3x3 matrix $A$ and a 3x1 vector $\mathbf{v}$, $A\mathbf{v}=\left(\begin{array}{c}{a}_{11}{v}_1+a_{12}{v}_2+a_{13}{v}_3\\ a_{21}{v}_1+a_{22}{v}_2+a_{23}{v}_3\\ a_{31}{v}_1+a_{32}{v}_2+a_{33}{v}_3\end{array}\right)$

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Strategy Tips

• Before the Exam: Understand each concept, not just memorize formulas.
• During the Exam:
  • Read each question carefully.
  • Plan your approach before writing.
  • Check if an answer makes sense in the context of the question.

Problem-Solving Approach

1. Identify: Determine what the problem is asking.
2. Plan: Choose which formulas or concepts apply.
3. Execute: Carry out your plan methodically.
4. Review: Check your work for errors or oversights.

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