Tags: #◇Computer Science #◇CSMATH2 #◇Linear Algebra #◇Reviewer #◇Finals

CSMATH2 Comprehensive Finals Reviewer

◇Overview

This reviewer consolidates Linear Algebra and Matrix Theory (CSMATH2). Core subjects include systems of linear equations, Gaussian elimination and Gauss-Jordan reduction to Reduced Row Echelon Form (RREF), matrix algebra and inverses, determinants, vector spaces, subspaces, linear independence, basis and dimension, linear transformations, eigenvalues, eigenvectors, and matrix diagonalization.

◇1. Master Matrix & Vector Theorem Matrix

Concept / Theorem	Mathematical Formulation	Invertibility Criteria / Condition
System Solutions	$A \mathbf{x} = \mathbf{b}$	Unique iff $\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) = n$ ; Infinitely many if rank $< n$ ; Inconsistent if $\text{rank}(A) < \text{rank}([A \mid \mathbf{b}])$ .
Matrix Invertibility	$A A^{-1} = A^{-1} A = I_n$	$A$ invertible $\iff \det(A) \ne 0 \iff \text{rank}(A) = n \iff \text{Nullity}(A) = 0$ .
Determinant Properties	$\det(A B) = \det(A)\det(B)$	$\det(A^T) = \det(A)$ , $\det(A^{-1}) = \frac{1}{\det(A)}$ , $\det(c A) = c^n \det(A)$ .
Rank-Nullity Theorem	$\text{rank}(A) + \text{nullity}(A) = n$	Where $n$ is the number of columns of $A$ .
Characteristic Equation	$\det(A - \lambda I) = 0$	Roots $\lambda$ are eigenvalues; $(A - \lambda I)\mathbf{v} = \mathbf{0}$ yields eigenvectors $\mathbf{v} \ne \mathbf{0}$ .
Diagonalization	$P^{-1} A P = D$	$A$ is diagonalizable $\iff A$ has $n$ linearly independent eigenvectors.

◇2. Row Operations & Determinant Effects

◇3. High-Yield Exam Checklist

Does $\det(A + B) = \det(A) + \det(B)$ ? NO! (Determinant does not distribute over addition).
For a scalar $c$ and $n \times n$ matrix $A$ , what is $\det(cA)$ ? $c^n \det(A)$ .
Is an eigenvector ever allowed to be the zero vector $\mathbf{0}$ ? NO, by definition $\mathbf{v} \ne \mathbf{0}$ .
How do you find the null space? Solve homogeneous system $A \mathbf{x} = \mathbf{0}$ .
Can a matrix with distinct eigenvalues always be diagonalized? YES, distinct eigenvalues guarantee linearly independent eigenvectors.

Previous Topic: ◇CSMATH1
Next Topic: ◇CSMODEL
Prerequisites: ◇Linear Algebra, Matrices, Vector Spaces, Eigenvalues
Cross-Curricular Connections: ◇CSINTSY

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