How to take linear algebra notes on iPad

Linear algebra often feels like two classes. The first few weeks are computation: row reduce a matrix, solve a system, multiply two matrices. Then, somewhere around vector spaces, it turns abstract. Suddenly the questions are "prove these vectors span R³" or "is this a subspace?", and students who coasted on computation hit a wall.

Notes that work for the whole course handle both halves. They keep computations neat enough to check, and they treat definitions and theorems as the core of the class, not as decoration around the examples.

Write row reduction so you can check it

Row reduction is long, repetitive and easy to get wrong in one entry. A single arithmetic slip in step two ruins everything after it. Neat, consistent layout is how you find that slip.

• Keep columns lined up. Each entry in a column should sit directly under the one above. Grid paper does this for you.
• Draw the augmented bar. A vertical line before the last column makes it clear which part is the coefficient matrix.
• Write the row operation on every step, like R2 → R2 − 3R1, next to the arrow between matrices. If an answer comes out wrong, you can redo each step in turn.
• One operation per step when you're learning. Combining several saves space but hides mistakes.
• Circle the pivots in the final echelon form. Pivot positions answer many questions at once: rank, which variables are free, whether columns are independent.

Grid paper is the natural choice for matrix work. In Scribba, you can change the paper of any page from the page sidebar, so lecture notes can stay ruled while computation pages use a grid. A landscape notebook gives a long chain of matrices room to run across the page; you choose orientation when you create the notebook.

Tip: check your final answer by plugging it back in. Multiply the original matrix by your solution vector and see if you get the right side. It takes a minute and catches most arithmetic errors.

Hold the Apple Pencil still at the end of a stroke to snap a straight line, which keeps matrix brackets and the augmented bar straight. For small subscripts like a₁₂ or v₃, the Zoom Window lets you write big in a magnified strip and lands your writing small and neat on the page.

Treat definitions as exact

In linear algebra, definitions are tools. To prove a set of vectors is linearly independent, you start from the definition of linear independence and work from there. If your version of the definition is fuzzy, you have nowhere to start.

Write each definition exactly as your professor or textbook states it, word for word. Then, underneath, add:

1. The idea in your own words. Linear independence means no vector in the set can be built from the others.
2. An example that meets the definition.
3. A non-example that fails it, and which part it fails.
4. How to check it in practice. For independence of vectors in Rⁿ, put them in a matrix and row reduce: independent if every column has a pivot.

Non-examples are especially valuable. Exams often ask whether something is a subspace, and the answer usually comes from showing one condition fails, like the set missing the zero vector.

Definitions make excellent flashcards. In Scribba, a Cornell Notes page has a CUES & QUESTIONS column beside the NOTES column, and Scribba reads your handwriting and turns each cue into a flashcard. Write the term as the cue ("span", "basis", "null space") and the exact definition beside it. Review and fix any misread words, then study the deck with spaced repetition. This doesn't need Apple Intelligence. See flashcards from handwritten notes.

Draw what the matrix does

Linear algebra has a geometric side that many courses rush through, and it's what makes the abstract half make sense. A 2×2 matrix is a transformation of the plane. Seeing that turns eigenvectors from a formula into something obvious.

Try these sketches on coordinate plane paper:

• Transform the unit square. Draw the square with corners at the origin, (1, 0), (0, 1) and (1, 1). Then draw where the matrix sends it. The columns of the matrix are where (1, 0) and (0, 1) land.
• Read the determinant. The area of the transformed square is the absolute value of the determinant. If the shape flips over, the determinant is negative. If it collapses to a line, the determinant is zero and the matrix isn't invertible.
• Find the eigenvectors. Look for directions that stay on their own line after the transformation. Those are eigenvectors, and how much they stretch is the eigenvalue.
• Draw a span. One nonzero vector spans a line; two independent vectors in R³ span a plane.

For 3D pictures of planes and vectors, isometric paper helps. See coordinate plane paper on iPad and isometric paper on iPad.

Keep one page for the big theorem

Most linear algebra courses build toward a single theorem that ties the course together, often called the Invertible Matrix Theorem. It lists many statements that are all equivalent for a square matrix: it's invertible, its determinant isn't zero, its columns are independent, its columns span Rⁿ, Ax = 0 has only the trivial solution, zero is not an eigenvalue, and more.

Give this theorem its own page and add each new condition as you learn it. Then you can answer a whole family of exam questions by connecting them: if the determinant is zero, you instantly know the columns are dependent. In Scribba, add a named bookmark to this page in the page sidebar so it's one tap away.

Practice proofs, not just computations

The abstract half of the course is usually graded on short proofs. A few habits help:

• Start from definitions. Write down what you're given and what you need to show, both in terms of definitions.
• Copy the structure of proofs from lecture. Many follow the same pattern, like checking the three subspace conditions in order.
• Re-prove theorems from class on a blank page without looking.

In Scribba, drag study tape over a proof or a worked computation, try it yourself on a fresh page, then tap the tape to peek. More in study tape on iPad.

Frequently asked questions

Can Scribba row reduce a matrix or check my answer?

No. Scribba doesn't recognize equations, do math or convert handwriting to LaTeX. It's for writing your work clearly and studying it later.

What paper is best for linear algebra?

Grid paper for row reduction and matrix arithmetic, coordinate plane paper for 2D transformations, and ruled paper for definitions and proofs.

How do I get better at linear algebra proofs?

Learn each definition word for word, then practice re-proving the theorems from lecture from a blank page. Most proofs start by writing down a definition.