Linear algebra assignment help should make the working visible. Markers want every row operation, every determinant step and every proof line, not just the final matrix.
This guide covers systems of equations, matrix algebra, determinants, vector spaces, eigenvalues and diagonalisation, with worked examples. It ends with how STEM Donkey prepares custom solutions for study and reference.
What Linear Algebra Markers Are Checking
Computation questions are marked on correct, clearly recorded steps, such as each row operation written as R2 → R2 − 2R1. Proof questions are marked on using the exact definitions: subspace tests, linear independence and the axioms of a vector space.
A final answer you can verify, by substituting back or multiplying out, earns trust and often partial credit.
Topics Linear Algebra Assignments Cover
| Topic | Typical questions | Key idea |
|---|---|---|
| Linear systems | Solve by Gaussian elimination, find all solutions | Row echelon form, pivots, free variables |
| Matrix algebra | Products, inverses, transposes | AB is generally not equal to BA |
| Determinants | Compute, use properties, test invertibility | det(A) ≠ 0 if and only if A is invertible |
| Vector spaces | Subspace tests, span, basis, dimension | Closure under addition and scalar multiplication |
| Linear maps | Kernel, image, matrix of a map | Rank-nullity theorem |
| Eigenvalues | Characteristic polynomial, eigenvectors, diagonalisation | Av = λv with v ≠ 0 |
| Inner products | Orthogonality, Gram-Schmidt, least squares | Projections onto subspaces |
Solving Linear Systems by Elimination
Gaussian elimination reduces the augmented matrix to row echelon form using three operations: swapping rows, scaling a row by a non-zero number, and adding a multiple of one row to another. None of these changes the solution set.
Worked example. Solve x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2.
R2 → R2 − 2R1 gives −3y − z = −9. R3 → R3 − R1 gives y − 2z = −4.
From the new third row, y = 2z − 4. Substituting into the second: −3(2z − 4) − z = −9, so −7z + 12 = −9 and z = 3. Then y = 2 and x = 6 − 2 − 3 = 1.
Check: 1 + 2 + 3 = 6, 2 − 2 + 3 = 3 and 1 + 4 − 3 = 2. All three hold.
A system has no solution if elimination produces a row such as 0 = 5, and infinitely many if there are fewer pivots than unknowns. In the second case, write the free variables as parameters and give the general solution.
Matrix Algebra and Determinants
Matrix multiplication is associative but not commutative. A product AB exists only when the number of columns of A equals the number of rows of B.
- 2 × 2 determinant: for rows (a, b) and (c, d), det = ad − bc.
- 2 × 2 inverse: swap a and d, negate b and c, and divide by ad − bc.
- Product rule: det(AB) = det(A)det(B).
- Row operations: swapping two rows flips the sign of the determinant; adding a multiple of one row to another leaves it unchanged.
- Transpose and inverse: (AB)T = BTAT and (AB)−1 = B−1A−1.
For 3 × 3 and larger matrices, row reduction to triangular form is usually faster than cofactor expansion. The determinant of a triangular matrix is the product of its diagonal entries.
Vector Spaces, Bases and Rank
To show a subset is a subspace, check three things: it contains the zero vector, it is closed under addition and it is closed under scalar multiplication. One failed condition with a specific counterexample shows it is not a subspace.
A basis is a linearly independent set that spans the space, and its size is the dimension. To test independence, set a linear combination equal to zero and show the only solution is all coefficients zero.
The rank-nullity theorem links the pieces: for an m × n matrix A, rank(A) + nullity(A) = n. If a 3 × 5 matrix has rank 3, its null space has dimension 2.
Eigenvalues, Eigenvectors and Diagonalisation
An eigenvalue λ of a square matrix A satisfies Av = λv for some non-zero vector v. Find eigenvalues from the characteristic equation det(A − λI) = 0, then solve (A − λI)v = 0 for each one.
Worked example. Let A have rows (2, 1) and (1, 2). Then det(A − λI) = (2 − λ)2 − 1 = λ2 − 4λ + 3 = (λ − 1)(λ − 3). The eigenvalues are 1 and 3.
For λ = 3: (A − 3I)v = 0 gives −v1 + v2 = 0, so v = (1, 1). For λ = 1: v1 + v2 = 0, so v = (1, −1).
Checks: the eigenvalues sum to 4, the trace of A, and multiply to 3, the determinant of A. A is symmetric, so its eigenvectors are orthogonal: (1, 1) · (1, −1) = 0.
A matrix is diagonalisable when it has n linearly independent eigenvectors. Then A = PDP−1, with the eigenvectors as the columns of P and the eigenvalues on the diagonal of D, in matching order. Real symmetric matrices are always diagonalisable by an orthogonal matrix.
Checking Your Answers in Code
Software cannot replace written working, but it is a quick way to check a hand calculation. A few lines of Python with NumPy confirm the example above.
import numpy as np A = np.array([[2, 1], [1, 2]]) vals, vecs = np.linalg.eig(A) print(vals) # 3 and 1 (order may vary) print(np.linalg.det(A)) # about 3.0 print(np.linalg.solve([[1, 1, 1], [2, -1, 1], [1, 2, -1]], [6, 3, 2])) # about [1. 2. 3.]
NumPy returns eigenvectors normalized to length 1 as the columns of vecs, so (1, 1) appears as about (0.707, 0.707). Scaled versions of an eigenvector are still eigenvectors, so both answers are correct.
Inner Products and Gram-Schmidt
Two vectors are orthogonal when their dot product is zero. The Gram-Schmidt process turns any basis into an orthogonal one by subtracting from each vector its projection onto the vectors already chosen.
Worked example. Start with u1 = (1, 1, 0) and u2 = (1, 0, 1). Keep v1 = u1.
The projection coefficient is (u2 · v1) / (v1 · v1) = 1 / 2. So v2 = (1, 0, 1) − ½(1, 1, 0) = (½, −½, 1).
Check: v1 · v2 = ½ − ½ + 0 = 0. Divide each vector by its length for an orthonormal basis.
Projections are also the idea behind least squares. The best approximation to b inside a subspace is its orthogonal projection, and the error vector is orthogonal to the whole subspace.
Where Linear Algebra Shows Up
- Least squares: the best-fit solution to an overdetermined system solves the normal equations ATA x = ATb.
- Differential equations: systems x' = Ax are solved with eigenvalues and eigenvectors.
- Data science: principal component analysis uses the eigenvectors of a covariance matrix.
- Engineering: stiffness matrices, circuit equations and vibration modes are all linear systems.
Common Linear Algebra Mistakes
| Mistake | Fix |
|---|---|
| Assuming AB = BA | Keep the order of products throughout |
| Not recording row operations | Write each one, such as R3 → R3 − R1 |
| Accepting the zero vector as an eigenvector | Eigenvectors must be non-zero |
| Proving a subspace with examples only | Show closure for arbitrary vectors and scalars |
| Mismatched order of P and D | Put each eigenvector in the same column as its eigenvalue |
How STEM Donkey Helps with Linear Algebra
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Frequently Asked Questions
Linear systems, matrix algebra, determinants, vector spaces, linear maps, eigenvalues, diagonalisation, inner products, Gram-Schmidt and least squares.
Put them as columns of a matrix and row reduce. They are independent if every column has a pivot; for a square matrix, a non-zero determinant shows the same.
No. A matrix needs n linearly independent eigenvectors. For example, the matrix with rows (1, 1) and (0, 1) has only one independent eigenvector, so it is not diagonalisable.
The number of pivots in its row echelon form, which equals the dimension of its column space and of its row space.
For checking, yes. Most courses still expect the characteristic polynomial and eigenvector working by hand for small matrices.
Yes. Ask for LaTeX in your brief and you receive the source file along with a PDF.
Yes, including proofs about subspaces, linear maps, kernels, isomorphisms and inner product spaces, written from the definitions your course uses.