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I am currently tutoring a student in linear algebra. She is a very hard worker and does well on computational problems, but struggles to build mathematical intuition. This struggle is compounded by the fact that she is not working in her native language.

Her attitude toward abstract concepts was resilient in the first month of the semester, but her determination seems to be slipping. Where she once worked to ask targeted questions, she's now resorted to the more typical "this doesn't make any sense."

In particular, she seems to have yet to realize the multitude of connections between bases, span, independence, dimension, rank, nullity, etc. She can get small pieces, but the big picture is definitely absent. True and false questions are a massive stumbling block.

I've done my best to provide visual illustrations for concepts where I can, but in what ways can I continue to support her in her efforts to learn the more abstract ideas? What sort of things should I encourage her to do in her own time? Should I tell her to deliberately spend time away from the material in the hope that we she returns she might have a better perspective?

I realize that this is a rather soft question, but any help whatsoever is much appreciated.

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Definitions and other facts

One thing I find particularly helpful with Linear Algebra is to help the student deal with the definitions in multiple ways. In Linear Algebra there are definitions, and there are properties that things have that are always true but aren't definitions. (They could have been chosen as definitions, but the chosen definition is more useful for proofs.) There are also things that are sometimes true. On top of this, there are definitions, and there are lists of things that are guaranteed to satisfy/not satisfy the definition.

For example, take linear independence/dependence.

The definition is that the list of vectors $\mathbf{v}_1$, ..., $\mathbf{v}_r$ is linearly independent when the only solution to the equation $x_1 \mathbf{v}_1 + ... + x_r \mathbf{v}_r = \mathbf{0}$ is $x_1=0$, ..., $x_r=0$. They are linearly dependent when there is at least one solution where at least one $x_i$ is not zero. This definition was chosen to make proofs and also numerical checking easier to do.

There is also the fact that the list of vectors $\mathbf{v}_1$, ..., $\mathbf{v}_r$ is linearly dependent when at least one of the vectors is a linear combination of at least one of the other vectors. This is perfectly true and makes more sense geometrically but it is not the definition. It is often easy to tell that the vectors are dependent using this idea, but it is not easy to tell they're independent using this idea.

Finally, there's a list of things that tell you that a set of vectors is linearly dependent/independent without directly using either of these. For example, if the list contains the zero vector, it's automatically dependent. If the list is made of vectors in $\mathbb{R}^n$ and it has more than $n$ vectors then it's automatically linearly dependent.

You can make a big table of these things with "linearly dependent" on one side and "linearly independent" on the other, highlighting things that are definitions and things that are always true and things that only work sometimes. I find this really helps students who are struggling with the multitude of ideas.

Similarly with subspaces. There is the definition with the three laws including closed under scalar multiplication and vector addition. But so often we don't describe what it looks like when something isn't a subspace. Showing some pictures and some equations that don't satisfy and why they don't is useful.

And then there are things you know must be subspaces: the span of any vectors and the solution to homogeneous linear equations. If the subspace is described by an equation, then if the equation isn't linear then it probably isn't a subspace. If the subspace describes the way to build the points, then if one of the linear combinations is missing it's not a subspace. A list of things that definitely are and things that definitely aren't is a great way of helping students come to terms with these things.

Sometimes they need a way of not always focussing on the abstract definition. They need a way of recognising the situation they're in without checking the definition, because when all is said and done, us experts don't check the definition every time either.

You can encourage her to make similar lists with this and other concepts in her own time. You can also encourage her to every time she sees a word, to attempt to remember the definition and several things that are true about it before doing anything else. This should help her remember the definition (which is important), but it should also give her something to do to stop her panicking.

Multiple representations and sets

I find a common issue is that students actually don't perceive the different ways of representing a subspace as different ways of representing the same thing. They also don't understand sets in general.

Most students I see struggling with linear algebra don't understand that the fundamental property of a set is a rule which tells you whether a thing is in the set or not in the set. It doesn't occur to them that they could check if a point is in a subspace or list points in the subspace. They don't see it that way. Thus the "closed under addition" thing is a bit mystical because they don't get the idea of "in".

I find it useful to draw out the similarities and differences between sets like this $\{(x,y,z) \in \mathbb{R}^3 | x+ 2y - z = 0\}$ and like this $\{(2a,3a-b,5b+a)|a,b \in \mathbb{R}\}$. The first explicitly lists a rule to check whether a point is in the set, but it's not easy to just list points that are in the set. The second which tells a way to construct all the points in the set easily, but it's not easy to tell if a random point is in the set. (The point of a basis, by the way, is to rewrite your subspace in the second "construct all the points" format.)

I would encourage her to, every time she is faced with a definition of a set, to make up some points and check if they are in it or not, and also to find several points in the set. Again, it will give her something to do to begin with before she gets overwhelmed. I'd also encourage her to look for multiple representations of the same thing. For example, to explicitly write, once a basis is found, something like $\{(x,y,z) \in \mathbb{R}^3 | x+ 2y - z = 0\}=\text{span }\{(1,0,1),(0,1,2)\}$

Proofs

The final issue is with proofs. Most students have not the slightest clue how to start them, or how to finish them, or how to write them. It's a style of maths writing and thinking they've never had to deal with before.

I find it very useful to look at existing proofs and put coloured circles around the different pieces to show that it has different parts doing different things. It's also useful to look at an example proof and reconstruct it from scratch, focussing in the decisions that were made at each stage of what to do. It's this deciding what to do that is the problem for most students. They're used to having methods to follow and making decisions for themselves is hard.

Finally, when doing the proof itself, I find it helps to write the information they have at the top of the page, and the goal they want to get to at the bottom. Then the proof goes in the middle. The major decision-making question that can be asked at any point they are stuck is "so what does this mean?". If your final line should be "$\mathbf{u}+\mathbf{v} \in V$", then this means that $\mathbf{u} + \mathbf{v}$ should satisfy whatever condition dictates whether a vector is in $V$. This may sound too simple, but believe me it is not.

The idea earlier of spending time investigating what it would mean to list a vector in the set and how you would check if a vector was in a set might help with this too. I could also recommend she looks through her notes herself and finds places where she thinks a decision had to be made so she can ask you/her teacher how the decision was made.

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    $\begingroup$ It is always shocking to me how many students seem unwilling or unable to translate $\{ (x+2y,2y-x,3x+7y) \ | \ x,y \in \mathbb{R} \}$ into a span. Sometimes, even $\{ a(x^2+1)+b(2x-3) \ | \ a,b \in \mathbb{R} \}$ is a mystery. I think your point that they don't understand basic set theory is on point. This is especially troubling in that many of them have had a proofs course before they take linear... $\endgroup$ – James S. Cook Oct 9 '16 at 23:48

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