How to study for a linear algebra midterm
Krsna Academy · · 6 min read
To study for a linear algebra midterm, learn how the big ideas connect, then practise explaining what each calculation tells you about the matrix. Row reducing quickly helps, but a midterm can also ask you to use a definition, justify a claim, or decide whether something is possible, and those questions need understanding that drills alone do not build.
What should you study first for a linear algebra midterm?
Start with your course outline, because the same subject can be taught quite differently. The University of Toronto's calendar describes MAT223H1 as a first course in R^n that emphasizes "the interplay between algebraic and geometric perspectives" and says it does not emphasize proofs. UBC's calendar lists systems of linear equations, matrix operations, determinants, eigenvalues and eigenvectors, and diagonalization of symmetric matrices for MATH 221. Other first courses, such as MATH 136 (Linear Algebra 1 for Honours Mathematics) at Waterloo or MATH*1160 (Linear Algebra I) at Guelph, may set different expectations again.
Make a list of the topics your midterm covers and sort it into three groups: what you can do without notes, what you can do with notes, and what you cannot do yet. Spend most of your time on the last two.
How do the main ideas in linear algebra connect?
Most of a first course circles one question: what does this matrix do to vectors, and can it be undone? For an n by n matrix A, the following statements are either all true or all false:
- A is invertible.
- The determinant of A is not zero.
- The rank of A is n.
- The columns of A are linearly independent and span R^n.
- The equation Ax = 0 has only the zero solution.
- Zero is not an eigenvalue of A.
Write this list from memory until you can explain why each line follows from another. If a question tells you that the determinant is zero, you immediately know five other facts.
How can one small example help you revise the whole course?
One two by two matrix can rehearse most of the midterm. Take the matrix with rows (1, 2) and (2, 4). The second column is twice the first, and here is what that one observation gives you:
| Question | Answer | What it means |
|---|---|---|
| Rank | 1 | The columns are dependent, so the matrix flattens the plane onto a line. |
| Column space | The multiples of (1, 2) | This line holds every possible output. |
| Null space | The multiples of (-2, 1) | These are the inputs sent to zero. Rank 1 plus nullity 1 equals 2 columns. |
| Determinant | 0 | The matrix cannot be inverted, because information is lost. |
| Eigenvalues | 0 and 5 | (-2, 1) goes to zero, and (1, 2) is stretched by 5. |
Now change one entry and predict what changes before calculating. Asking "what if" this way builds the judgment that method drills skip.
What mistakes cost the most marks in linear algebra?
The costliest mistakes tend to be misunderstandings of what an operation preserves, not slips in arithmetic. Watch for these:
- Taking a column space basis from the reduced matrix. Row operations can change the column space, so use the pivot positions to choose columns from the original matrix.
- Reading eigenvalues from a row reduced matrix. Row operations generally change eigenvalues, so work from the original matrix.
- Forgetting how row operations affect the determinant. A swap flips the sign, scaling a row scales the determinant, and adding a multiple of one row to another changes nothing.
- Treating matrix multiplication like ordinary multiplication. AB and BA usually differ, and the inverse of AB is the inverse of B times the inverse of A.
- Confusing invertible with diagonalizable. The example above is diagonalizable but not invertible, while the matrix with rows (1, 1) and (0, 1) is invertible but not diagonalizable.
- Calling a set a subspace without checking it. A subspace must contain the zero vector and stay closed under addition and scalar multiplication, so a line that misses the origin fails.
What is a good one-week study plan?
A week is enough to make real progress if you practise actively each day. Adjust this plan to your own midterm date and syllabus:
- Day 1: Sort your topics into the three groups, and write every definition from memory.
- Day 2: Practise row reduction, solving systems, and describing solution sets in words.
- Day 3: Work on span, independence, basis, and dimension, and draw examples in two and three dimensions.
- Day 4: Work on linear transformations, rank, and null space, using the small matrix method above.
- Day 5: Practise determinants, eigenvalues, and diagonalization if your midterm covers them.
- Day 6: Write a past midterm under timed conditions, then mark it honestly.
- Day 7: Redo only what went wrong, and review your definitions and the list of connected statements.
Check answers as you go. Multiply your eigenvector by the matrix, substitute your solution into the original system, and test whether your rank and nullity add up to the number of columns. These checks take seconds and are worth using during the midterm itself.
When does a linear algebra tutor help?
A tutor helps most when you can follow the lectures but cannot start an unfamiliar question, or when proofs and definitions still feel like a foreign language. Krsna Academy offers one-to-one linear algebra tutoring built around your course outline and past tests, including MAT223 at UofT and MATH*1160 at Guelph. If your midterm is only days away, see our university exam tutoring.
Questions people ask
How do I study for a linear algebra midterm if I only have a few days?
Prioritize what your outline says is on the midterm, then learn the definitions and the connected facts about invertibility, rank, and determinants. Spend the rest of your time on a past midterm under timed conditions and redo every question you missed. Practising unfamiliar questions usually helps more than rereading notes.
Why is linear algebra so hard when the calculations are easy?
The calculations are often simple, but the course also asks what they mean. Span, basis, and subspace are abstract ideas, and many students meet proofs or precise definitions at this level for the first time. Picturing the ideas in two and three dimensions first makes the general definitions easier to use.
Do I need to memorize proofs for a linear algebra midterm?
It depends on the course, so check your outline and past tests. The University of Toronto's calendar, for example, says MAT223H1 does not emphasize proofs, while a proof-based course may expect much more. In either case, knowing the exact definitions is the best place to start.
What is the fastest way to check an eigenvector?
Multiply the matrix by your vector and see whether the result is a scalar multiple of that vector. That scalar is the eigenvalue. The vector must also be nonzero, although an eigenvalue can be zero.
Written by Krsna Academy, a one-to-one tutoring academy for learners from Grade 1 to university across Canada. Krsna Academy was founded by Mehak Saini, who holds a mathematics degree from Dalhousie University and taught IB mathematics, physics, and economics for six years.