This page covers Linear Algebra at the College Freshman level, delivered as a formula cheat sheet. Vectors, matrices, determinants, eigenvalues, and linear transformations. The mathematics of data sc. The material here corresponds to First year of university courses: Calculus 1 and Calculus 2.
The key formulas for Linear Algebra at the College Freshman level are organised below. Each formula is accompanied by a note on when it applies and what common variations exist.
The skills covered by these formulas are: Matrix operations, Gaussian elimination, Determinants, Eigenvalues and eigenvectors, Linear transformations.
For each formula, read the conditions carefully. Many errors in Linear Algebra come from applying a formula outside its domain of validity — using a geometric formula that assumes a right angle when the angle is not specified, or applying a probability rule that requires independence when the events are dependent.
Use this sheet as a revision tool after you have worked through problems — not as a first introduction to the material. A formula you have derived or used is one you will remember; a formula you have only read is one you will forget under exam pressure.
Worked Example
A standard linear algebra problem at the college freshman level.
Work through step by step: identify what is given, what is asked, apply the relevant technique, and check your answer against the original conditions.
Assuming matrix multiplication is commutative: AB ≠ BA in general. Matrix multiplication is commutative only for special cases (scalar multiples, diagonal matrices, etc.).
Frequently Asked Questions
How is Linear Algebra different at the College 1st Year level compared to earlier levels?
At the College Freshman level, Linear Algebra builds on First year of university prerequisites. Students are expected to have completed Calculus 1 before tackling this material.
Which exams test Linear Algebra at this level?
GRE Math, Engineering boards, Graduate school prerequisites.
What is the single most effective way to practise Linear Algebra for College 1st Year students?
The most effective practice at the College Freshman level is deliberate work on novel problem setups — not repeated drilling of the same template. Attempt problems before looking at solutions, and review errors by identifying the specific step where the reasoning broke down.