26 topics
Linear algebra help, from row reduction to eigenvectors
Linear algebra is the course where the arithmetic is easy and the vocabulary is not. You can row reduce a matrix perfectly and still be unable to say what the span of its columns is, or why a determinant of zero means the system has either no solution or infinitely many. Here the definitions are the subject, and the computation is the part that already works.
Where students get stuck
My row reduction gives a different matrix than the answer key
Both of you can be right. Row echelon form is not unique — it depends on which multiples you used and in what order — so two correct people finish with different matrices. Reduced row echelon form is unique, so reduce all the way if you want to compare directly. What actually has to match is the pivot positions and the solution set, not the entries. If your pivots sit in the same columns and your parametric solution describes the same set of vectors, you did it right.
I can compute a determinant but I don’t know what it tells me
The determinant is a signed volume scaling factor. Push the unit square, or the unit cube, through the matrix and the determinant is the area or volume of what comes out, with a minus sign if the orientation flipped. That one fact explains the rules you memorised. A determinant of zero means the matrix squashes space into a lower dimension, so information is destroyed, so the map cannot be undone — which is exactly why zero determinant, not invertible, dependent columns, and a system with no solutions or infinitely many are all the same statement.
Span, linear independence and basis all sound like the same thing
They answer three different questions about the same set of vectors. Span asks how much you can reach: everything you can build by scaling them and adding them together. Linear independence asks whether any of them is redundant: can one be built out of the others. A basis is the set that is both — it reaches the whole space and carries nothing spare. Say it as enough and not too many. A basis is exactly enough, which is why every basis of a given space has the same number of vectors, and why that number is called the dimension.
I can find eigenvalues but I have no idea what an eigenvector is for
An eigenvector is a direction the matrix refuses to turn. Multiply it and the result points the same way, only stretched by the eigenvalue. Almost every other vector gets rotated, so these are the natural axes of that particular transformation. Written in those directions, the matrix behaves like plain multiplication by numbers — that is all diagonalisation is doing. It is also why anything repeated, such as a matrix power or a system stepping forward in time, suddenly becomes cheap: you raise a few numbers to a power instead of multiplying matrices over and over.
What's covered
Linear Algebra topics you can work through with a tutor, generate practice on, or turn into flashcards and a study plan.
Systems and matrices
- Gaussian elimination and row echelon form
- Reduced row echelon form and pivot columns
- Matrix addition, multiplication and transpose
- Inverses and the invertible matrix theorem
- LU factorisation
Vectors and vector spaces
- Linear combinations and span
- Linear independence
- Subspaces, basis and dimension
- Column space, row space and null space
- Rank and the rank–nullity theorem
Determinants
- Cofactor expansion
- Determinants by row reduction
- How row operations change the determinant
- Cramer’s rule
- Volume, orientation and singular matrices
Eigenvalues and diagonalisation
- The characteristic polynomial
- Eigenvalues, eigenvectors and eigenspaces
- Algebraic and geometric multiplicity
- Diagonalisation and powers of a matrix
- Symmetric matrices and orthogonal diagonalisation
Orthogonality and applications
- Dot product, length and angle
- Orthogonal projection onto a subspace
- The Gram–Schmidt process
- Least squares and line of best fit
- Linear transformations and their matrices
- Change of basis
Linear Algebra questions
Is this the matrix work from high school, or the first-year university course?
The topic list above is the university course — proofs about subspaces, rank, diagonalisation. High school matrix work sits inside the first two groups. Say which one you are in and it will pitch the notation and the amount of justification at that level rather than defaulting to the harder one.
Can it show me what a transformation actually does to a shape?
Yes. On the whiteboard it will draw a square or a set of basis arrows, apply the matrix, and sketch the result beside it. That picture is the fastest way to make shear, projection, rotation and a zero determinant stop being definitions and start being things you can see.
Will it just do my problem set for me?
It will not hand you a finished assignment. What it does is work a matching problem with different numbers while you watch, then hand yours back to you a step at a time and check each line. If you paste in a full question and ask for the answer, expect to be walked through it instead.
Can I drill one narrow thing, like Gram–Schmidt on its own?
Yes. Ask for a quiz on a single technique and you get a fresh set of problems on only that, each with the full solution underneath. People usually use this the night before a test on whichever one topic keeps going wrong.
Stuck on linear algebra right now?
Talk it through out loud, share your screen, and watch it worked out step by step on a whiteboard.
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