Linear algebra is less about finding the “final answer”, and more about explaining why the answer is accurate. Here are a few tips that will help you improve your full solutions so that they clearly communicate what you’re trying to say:

*Avoid using pronouns unless it’s clear what they refer to.*“It” could be a matrix, or an equation, or a vector… if in doubt, just use the word.*Don’t write \(=\) unless two things are actually equal.**Almost never write \(\rightarrow\) in an explanation.*Sometimes the arrow should be an equal sign. Other times the arrow should be a sentence explaining what that “next step” is doing. It’s okay to use an arrow if you’re actually just using it to point out something.*When using a matrix, explain how and why you constructed it.*Often, you need a matrix because it represents a system of equations - describe that system.*Use “consistent” and “inconsistent” correctly.*A matrix or vector cannot be consistent or inconsistent. But a system of equations or a vector equation can be consistent (has a non-empty solution set) or inconsistent (has an empty solution set due to a contradicition, usually \(0=1\)). It’s sometimes okay to say “the system of equations represented by this augmented matrix is (in)consistent” to avoid writing out the system, but notice that the matrix must be augmented for this to make sense.*Don’t just point out the feature of an RREF matrix.*Many problems require pointing out a feature of an RREF matrix, but you need to explain why that feature is relevant to the exercise.*Answer the question asked.*After your explanation, be sure to conclude with the (often) “yes” or “no” answer.*Don’t write a sentence in place of an established mathematical symbol.*For example, “we now use technology to compute the RREF of the matrix A to result in the matrix B” could be much more quickly communicated by just writing \(\operatorname{RREF}(A)=B\) or \(A\sim B\).

The sample solutions PDF will help you find the right words, but it’s important you know what those words mean: it’s very easy for me to tell when a student has memorized a sentence without understanding it most of the time, because the sentence they wrote usually isn’t exactly what I said (and no longer makes any sense because of the error). Students who know what they’re trying to say won’t write exactly what I put in the sample solution, but they’ll write something else with the same meaning.

These tips are specific to our linear algebra course, but for general advice on how to clearly communicate mathematics, Prof. Francis Su has written a great overview. To see what “professional” mathematical writing looks like, here’s an example of a research paper I’ve written. Or just look in your favorite textbook.

## Tips on explaining spanning/independence

We’ve covered lots of ways to tell if a set of vectors is spanning
or independent, but here the basic definitions I’d appeal to in order
to *explain* why a set of vectors is spanning or linearly indepednent (since they
directly involve vector equations you can solve with an augmented matrix).

- A set \(\{\vec v_1,\dots,\vec v_n\}\) of vectors spans a space \(V\) provided that for each \(\vec v\in V\), there is at least one solution to the vector equation \(x_1\vec v_1+\dots+x_n\vec v_n = \vec v\).
- A set \(\{\vec v_1,\dots,\vec v_n\}\) of Euclidean vectors is linearly independent provided that the only solution to the vector equation \(x_1\vec v_1+\dots+x_n\vec v_n = \vec 0\) is \(x_1=\dots=x_n=0\).

If you need to show something about spanning or independence to solve a larger problem, then feel free to just say something about pivots, rows, and/or columns.