15 Common Mistakes When Studying Linear Algebra (And How to Fix Them) | LearnByTeaching.ai
Linear algebra is a foundational course for mathematics, engineering, and data science, but its abstract nature trips up even strong students. Many learners can row-reduce matrices mechanically yet struggle with the geometric and conceptual meaning behind the computations. Here are 15 common mistakes and how to fix them.
Treating matrix multiplication as element-wise
Students who are comfortable with scalar multiplication assume AB means multiplying corresponding entries. Matrix multiplication is defined as dot products of rows and columns, and AB is not the same as BA.
A student multiplies two 2x2 matrices entry-by-entry instead of taking the dot product of each row of A with each column of B, getting a completely wrong result.
How to fix it
Always think of matrix multiplication as 'row dot column.' Write out the dot products explicitly until the operation becomes automatic. Verify by checking dimensions: an m x n times n x p gives m x p.
Confusing rank with determinant
Students think a nonzero determinant and full rank are interchangeable concepts in all cases, leading to errors with non-square matrices where determinants don't exist.
A student tries to compute the determinant of a 3x5 matrix to determine its rank, not realizing determinants only apply to square matrices.
How to fix it
Remember that rank is defined for any matrix (number of linearly independent rows or columns), while determinant only exists for square matrices. Use row reduction to find rank universally.
Ignoring the geometric meaning of eigenvectors
Students compute eigenvalues and eigenvectors purely algebraically without understanding that eigenvectors are the directions a linear transformation stretches or compresses.
A student correctly finds eigenvalue 2 with eigenvector [1, 1] but cannot explain that this means vectors along y = x get doubled in length by the transformation.
How to fix it
For every eigenvector problem, sketch the transformation in 2D. Watch 3Blue1Brown's Essence of Linear Algebra chapter on eigenvalues to build geometric intuition.
Forgetting that row operations change the determinant
When row-reducing to find a determinant, students forget that swapping rows flips the sign and scaling a row multiplies the determinant.
A student row-reduces a matrix to upper triangular form, multiplies the diagonal entries, and reports the product as the determinant without accounting for two row swaps that should flip the sign twice.
How to fix it
Keep a running tally of row operations: each swap multiplies det by -1, scaling row by k multiplies det by k, and adding a multiple of one row to another doesn't change det.
Assuming all linear transformations are invertible
Students default to thinking every matrix has an inverse, leading to errors when they try to 'divide' by a singular matrix.
Solving AX = B, a student writes X = A^(-1)B without checking whether A is invertible, producing nonsense when det(A) = 0.
How to fix it
Before inverting any matrix, check the determinant or verify the matrix has full rank. If singular, use the general solution approach with row reduction instead.
Confusing the null space with the column space
These two fundamental subspaces answer different questions (what maps to zero vs. what can be output), but students blend them together when the notation looks similar.
Asked to find all solutions to Ax = 0, a student computes the column space of A instead of the null space.
How to fix it
Memorize the definitions: null space = {x : Ax = 0}, column space = {b : Ax = b has a solution}. They live in different vector spaces (R^n vs R^m for an m x n matrix).
Skipping the proof that a set is a subspace
Students identify subspaces by gut feeling rather than checking the three requirements: contains zero vector, closed under addition, closed under scalar multiplication.
A student claims {(x, y) : x + y = 1} is a subspace of R^2, not noticing it doesn't contain the zero vector.
How to fix it
Always check all three subspace conditions. A quick sanity check: does the set contain the zero vector? If not, it's immediately not a subspace.
Memorizing formulas instead of understanding the algorithm
Students memorize the 2x2 inverse formula but have no strategy for larger matrices, because they never learned the row-reduction-based inversion method.
A student can invert a 2x2 matrix using ad-bc but is completely lost when asked to invert a 3x3 matrix on an exam.
How to fix it
Learn to invert any matrix by augmenting [A | I] and row-reducing to [I | A^(-1)]. The formula is a shortcut; the algorithm is the real tool.
Not understanding change of basis
Students can compute in the standard basis but are confused when a problem uses a different basis, because they don't realize the same vector or transformation looks different in different coordinates.
Given transformation matrix T in basis B, a student applies T directly to standard-basis coordinates without converting, getting the wrong answer.
How to fix it
Think of change of basis as a translation step: convert to the working basis, apply the transformation, convert back. Practice with explicit 2D examples using non-standard bases.
Cramming before exams instead of practicing problems daily
Linear algebra builds cumulatively, and last-minute studying leaves no time to internalize the connections between concepts like rank, invertibility, determinants, and eigenvalues.
A student rereads notes the night before the exam but can't solve novel problems that combine multiple concepts, like showing a matrix is diagonalizable.
How to fix it
Do 3-5 problems daily throughout the course. Each concept connects to many others, and the only way to see those connections is sustained practice.
Confusing linear independence with orthogonality
Students think linearly independent vectors must be perpendicular, or that non-orthogonal vectors are dependent.
A student claims [1, 0] and [1, 1] are linearly dependent because they are not perpendicular.
How to fix it
Linear independence means no vector in the set can be written as a linear combination of the others. Orthogonality is a stronger condition. Draw examples: [1, 0] and [1, 1] are independent but not orthogonal.
Errors in computing the characteristic polynomial
Sign errors and expansion mistakes when computing det(A - lambda*I) are extremely common, especially for 3x3 or larger matrices.
A student expands a 3x3 characteristic polynomial but misses a sign in the cofactor expansion, getting wrong eigenvalues.
How to fix it
Double-check by substituting your eigenvalues back into det(A - lambda*I) to verify you get zero. Use cofactor expansion along the row or column with the most zeros.
Not verifying solutions to Ax = b
After solving a system, students submit the answer without plugging it back into the original equations to check.
A student makes an arithmetic error during row reduction and reports an incorrect solution that doesn't satisfy the second equation.
How to fix it
Always substitute your solution vector back into the original system. This 30-second check catches most arithmetic errors on exams.
Misunderstanding what 'span' means
Students confuse the span of a set of vectors with the set itself, not realizing span means all possible linear combinations.
A student says span{[1, 0], [0, 1]} contains only two vectors rather than recognizing it is all of R^2.
How to fix it
Span{v1, ..., vk} = {c1*v1 + ... + ck*vk for all scalars c1, ..., ck}. It's typically an infinite set. Practice by describing the span of small vector sets geometrically.
Rushing through the diagonalization process
Students forget to check whether a matrix is diagonalizable before attempting to diagonalize it, or they assemble the matrices P and D in the wrong order.
A student finds two eigenvalues for a 3x3 matrix but only two linearly independent eigenvectors, then gets stuck because the matrix isn't diagonalizable.
How to fix it
Before attempting diagonalization, verify that you have n linearly independent eigenvectors for an n x n matrix. Remember the factorization A = PDP^(-1) where columns of P are eigenvectors in the same order as eigenvalues on the diagonal of D.
Quick Self-Check
- Can you explain what rank means geometrically, not just how to compute it?
- Do you know when a matrix is invertible using at least three equivalent conditions?
- Can you describe what an eigenvector represents without mentioning the computation?
- Can you determine whether a set of vectors is a subspace by checking all three conditions?
- Can you set up and solve a change-of-basis problem from scratch?
Pro Tips
- ✓Build a master equivalence list: A is invertible if and only if det(A) != 0, rank(A) = n, null space is trivial, columns are linearly independent, eigenvalues are all nonzero. Knowing these connections is the key to linear algebra.
- ✓Use 3Blue1Brown's 'Essence of Linear Algebra' series to build geometric intuition before diving into proofs and computation.
- ✓For every abstract concept, construct a concrete 2x2 or 3x3 example. If you can't illustrate it with a small matrix, you don't understand it yet.
- ✓Connect each concept to an application: PCA for data science, PageRank as an eigenvector problem, least squares for statistics.
- ✓When studying for exams, focus on problems that combine multiple concepts (e.g., 'show this matrix is not diagonalizable') rather than isolated computations.