Chegg Linear Algebra — Free Alternatives, Topics & Study Resources

Chegg is a paid online study platform that provides the following for linear algebra:

1. Textbook Solutions: • Step-by-step solutions to popular linear algebra textbooks • Covers books like: Linear Algebra and Its Applications (David C. Lay), Introduction to Linear Algebra (Gilbert Strang), Elementary Linear Algebra (Howard Anton) • Solutions for odd and even numbered problems

2. Expert Q&A: • Submit a linear algebra question and get an answer from a subject expert • Typically answered within a few hours • Includes step-by-step explanations

3. Chegg Study: • Monthly subscription (~$14.95/month as of 2026) • Access to millions of textbook solutions across subjects • Practice problems with guided solutions

Limitations of Chegg: • Requires paid subscription • Answers may contain errors (user-submitted) • Risk of academic integrity violations if used to copy homework • Does not help build understanding — just provides answers • Some students become over-dependent on it

Important: Using Chegg to copy answers on graded assignments may violate your university's academic honesty policy. Use it as a learning tool, not a shortcut.

These free resources are excellent for learning linear algebra without a Chegg subscription:

1. MIT OpenCourseWare (OCW) — ocw.mit.edu • Gilbert Strang's legendary 18.06 Linear Algebra course • Full video lectures, notes, problem sets, and exams with solutions • Completely free — considered the gold standard

2. Khan Academy — khanacademy.org • Linear algebra course covering vectors, matrices, transformations • Interactive exercises with instant feedback • Free with no subscription required

3. 3Blue1Brown (YouTube) — 'Essence of Linear Algebra' • Beautiful visual explanations of linear algebra concepts • Covers vectors, linear transformations, determinants, eigenvalues • Best for building geometric intuition

4. Paul's Online Math Notes — tutorial.math.lamar.edu • Detailed notes with worked examples • Covers systems of equations, matrices, determinants, eigenvalues

5. Wolfram Alpha — wolframalpha.com • Free step-by-step solutions for matrix operations • Can compute determinants, eigenvalues, row reduction, etc. • Partial free access; full steps require Pro

6. Symbolab — symbolab.com • Free matrix calculator with step-by-step solutions • Handles row echelon form, inverses, eigenvalues

7. Professor Leonard (YouTube) • Full-length university linear algebra lectures • Clear, thorough teaching style

8. Shinyu.ai • AI-powered study tools for generating quizzes, flashcards, and summaries • Upload your linear algebra notes for personalised study materials

Here are all the essential topics covered in a typical linear algebra course:

1. Systems of Linear Equations: • Solving by substitution, elimination, and matrix methods • Gaussian elimination and row echelon form • Reduced row echelon form (RREF) • Homogeneous and non-homogeneous systems • Consistent vs inconsistent systems

2. Matrices: • Matrix addition, subtraction, scalar multiplication • Matrix multiplication (AB ≠ BA in general) • Transpose (Aᵀ), Symmetric matrices • Inverse of a matrix (A⁻¹), conditions for invertibility • Elementary row operations • Identity matrix, zero matrix

3. Determinants: • Determinant of 2×2 and 3×3 matrices • Cofactor expansion (Laplace expansion) • Properties: det(AB) = det(A)·det(B), det(Aᵀ) = det(A) • Cramer's Rule for solving systems • A matrix is invertible if and only if det(A) ≠ 0

4. Vector Spaces: • Vectors in Rⁿ, vector addition, scalar multiplication • Subspaces, span, linear independence • Basis and dimension • Row space, column space, null space • Rank-Nullity Theorem: rank(A) + nullity(A) = n

5. Linear Transformations: • Definition: T(au + bv) = aT(u) + bT(v) • Matrix representation of transformations • Kernel (null space) and range (image) • One-to-one and onto transformations

6. Eigenvalues and Eigenvectors: • Characteristic equation: det(A − λI) = 0 • Finding eigenvalues and eigenvectors • Diagonalisation: A = PDP⁻¹ • Applications: differential equations, Markov chains, Google PageRank

7. Orthogonality: • Dot product, length, angle between vectors • Orthogonal and orthonormal sets • Gram-Schmidt process • Orthogonal projections • Least squares solutions

Problem 1: Solve the system using Gaussian elimination 2x + y − z = 8 −3x − y + 2z = −11 −2x + y + 2z = −3

Solution: Augmented matrix: [2 1 −1 | 8; −3 −1 2 | −11; −2 1 2 | −3] R2 → R2 + (3/2)R1: [2 1 −1 | 8; 0 1/2 1/2 | 1; −2 1 2 | −3] R3 → R3 + R1: [2 1 −1 | 8; 0 1/2 1/2 | 1; 0 2 1 | 5] R3 → R3 − 4R2: [2 1 −1 | 8; 0 1/2 1/2 | 1; 0 0 −1 | 1] Back substitution: z = −1, y = (1 − 1/2(−1))/(1/2) = 3, x = (8 − 3 + (−1))/2 = 2 Answer: x = 2, y = 3, z = −1

Problem 2: Find the determinant of A = [3 1 2; 0 4 −1; 2 5 3] Solution: Expand along row 1: det(A) = 3(4·3 − (−1)·5) − 1(0·3 − (−1)·2) + 2(0·5 − 4·2) = 3(12 + 5) − 1(0 + 2) + 2(0 − 8) = 3(17) − 1(2) + 2(−8) = 51 − 2 − 16 = 33

Problem 3: Find eigenvalues of A = [4 1; 2 3] Solution: det(A − λI) = 0 (4 − λ)(3 − λ) − (1)(2) = 0 12 − 7λ + λ² − 2 = 0 λ² − 7λ + 10 = 0 (λ − 5)(λ − 2) = 0 Eigenvalues: λ₁ = 5, λ₂ = 2

Essential formulas every linear algebra student must know:

Matrices: • (AB)ᵀ = BᵀAᵀ • (AB)⁻¹ = B⁻¹A⁻¹ • (Aᵀ)⁻¹ = (A⁻¹)ᵀ • A · A⁻¹ = A⁻¹ · A = I (identity matrix)

2×2 Matrix Inverse: • If A = [a b; c d], then A⁻¹ = (1/det(A)) · [d −b; −c a] • det(A) = ad − bc

Determinant Properties: • det(AB) = det(A) · det(B) • det(Aᵀ) = det(A) • det(kA) = kⁿ · det(A) for n×n matrix • det(A⁻¹) = 1/det(A) • Swapping two rows changes the sign of det

Vector Spaces: • Rank-Nullity Theorem: rank(A) + nullity(A) = number of columns • dim(Row space) = dim(Column space) = rank(A) • dim(Null space) = nullity(A)

Eigenvalues: • Characteristic equation: det(A − λI) = 0 • Sum of eigenvalues = trace(A) = a₁₁ + a₂₂ + ... + aₙₙ • Product of eigenvalues = det(A) • If A is triangular, eigenvalues = diagonal entries

Orthogonality: • Dot product: u · v = u₁v₁ + u₂v₂ + ... + uₙvₙ • ||u|| = √(u · u) • Projection of u onto v: proj_v(u) = (u · v / v · v) · v • u and v are orthogonal if u · v = 0

Linear algebra is conceptual — memorising formulas alone is not enough. Here are proven study strategies:

1. Build Geometric Intuition: • Watch 3Blue1Brown's 'Essence of Linear Algebra' series before starting the course • Visualise vectors, transformations, and spaces — do not just compute • Understand WHAT a matrix does (transforms space) not just HOW to multiply

2. Practice Row Reduction Until It's Automatic: • Gaussian elimination is the foundation of linear algebra • Practice until you can do RREF quickly and without errors • Most problems reduce to row reduction at some point

3. Understand Definitions Deeply: • Linear independence, span, basis, dimension — know the precise definitions • Most proofs and problems come directly from definitions • Ask: 'What does this definition MEAN geometrically?'

4. Connect Topics: • Invertible Matrix Theorem: A is invertible ↔ det(A) ≠ 0 ↔ rank = n ↔ null space = {0} ↔ columns are linearly independent ↔ eigenvalues are all non-zero • These are ALL the same statement from different perspectives

5. Do Problems by Hand: • Do not rely on calculators or Chegg for homework • Working through computations builds understanding • Use tools to CHECK your work, not to DO your work

6. Form Study Groups: • Explaining concepts to others deepens your understanding • Linear algebra has many 'aha!' moments that come from discussion

7. Use Multiple Resources: • MIT OCW for lectures, Khan Academy for practice, 3Blue1Brown for intuition • No single resource is perfect — combine them