Learning the Language of Linear Algebra John Hannah (Canterbury, NZ) Sepideh Stewart (Oklahoma, US) Mike Thomas (Auckland, NZ)

Summary Goals for a linear algebra course Experiments and writing tasks Examples Student views

What do today’s students need? Algorithms – Gaussian elimination, Gram-Schmidt process Concepts – Matrix factorizations, projections Language – Span, linear independence, basis, dimension Communication skills – Reaching the lay audience

Learning mathematics How do we as mathematicians learn math? Given some beginning stimulus - perhaps a talk we listen to, or a paper we read, or just a thought we have - we do examples, make conjectures, solve problems, prove theorems, and communicate with our colleagues, both in talking and writing. David Carlson, MAA Invited Address, 1992

Doing mathematics We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics. William Thurston, Bull. AMS, 1994

How adults learn (Jarvis)

Experiments and write-ups Lab sessions allow us to – use MATLAB, avoiding distracting details of long calculations – generate rich experiences involving many examples – give scope for forming and testing conjectures – set writing tasks calling for reflection and synthesis.

The lab melting pot

Typical writing task Write a short report describing your results Use complete English sentences (with few or no symbols or equations) Any classmate should be able to read and understand your report

Example 1 Let u 1, u 2, u 3 be vectors in 2-space. Does u 3 usually belong to the span of u 1, u 2 ? Are 3 vectors in 2-space usually linearly independent? Interpret your results geometrically.

Sample response 1 Given three vectors u 1, u 2 and u 3 in 2-space, u 3 is usually in the span of u 1 and u 2. This is because it is impossible for u 1, u 2 and u 3 to all be linearly independent in 2-space as any two linearly independent vectors would span all of 2-space, thus u 3 would be a linear combination of u 1 and u 2, and thus in the span of u 1 and u 2 …

Example 2 Let A be a random 1x5 matrix. How many parameters in the solution to Ax=0? For which b is Ax=b solvable? How many independent equations in Ax=0? Find the dimensions of the solution, column and row spaces of A. What usually happens if you enlarge A by adding another random row? and another? Are there any relationships between the dimensions you found above?

Sample response 2 (row space) When a random 2x5 matrix B is created, and we have a system Bx=0, we have a system of 2 equations in 5 unknowns, and it is very unlikely that the 2 rows are linearly dependent, so typically there are 2 independent equations and so the dimension of the row space is 2.

Sample response 2 (solution space) Examining the reduced row echelon form of B, there are typically 3 parameters to the general solution of Bx=0. This means there are 3 vectors spanning the solution space, they are most likely linearly independent, so the basis of the solution space consists of 3 vectors, and the dimension of the solution space is 3.

Sample response 2 (column space) Vectors b for which the system Bx=b are consistent are those which belong to the column space of B. As the columns represent 5 vectors in 2-space, it most likely that there are two linearly independent columns, and the dimension of the column space is 2, meaning there is a 2 dimensional space of b’s that make Bx=b consistent.

Sample response 2: synthesis From the previous typical cases it can be seen that the dimension of the row space plus the dimension of the null space is equal to the number of columns in the matrix. Given that there are m equations in the system, it is typically the case that a consistent right-hand side b can be any vector in m-space provided all m rows of coefficient matrix are linearly independent.

Example 3 Consider the vectors u=(1,0,0), v=(0,2,0) and w=(3,4,0). Write a short paragraph about u, v and w. Your paragraph should be at most 75 words long, but should include as many as possible of the following technical terms from Linear Algebra: basis, dimension, dependence relation, linear combination, linearly dependent, linearly independent, span, subspace.

Student C does Example 3 u, v, w form a subspace in R 3 because they can contain the zero vector, and are closed in addition and multiplication. … If these vectors formed a matrix the dimension of the matrix would be the basis, or number of linear independent rows. Which could be found by row reducing.

Student C on labs Yeah, because we had to work in groups, and that was a bad part about it as well, because it required a lot more in-depth reading. You couldn’t just briefly scan it and know it, and therefore it wasn’t really good for working in groups, because that’s what you have a tendency to do, is, kind of, you actually get more in-depth with it, I think.

Student F I kind of like the tutorials we were doing, we were actually writing about, um, the concepts as opposed to just doing it, small problems, … ‘cos they would be just really small problems, like 30 or 40 of them for a tutorial, which always seemed to me like I wasn’t achieving much. And then actually trying to think about … how to explain something really helped improve my understanding of those topics.

Student A I thought it was good. Some of it is quite tough, communicating with your partner, because we did the labs in pairs and, just a few, if you think you’ve got it right, but you don’t know how to explain it, it’s really hard to communicate it to your partner, especially if you’ve got it wrong.

Conclusion (Student H) The tutorial structure also helped with that a lot, having to write out reports in which we had to use all this language and, yeah, it was a case of having to use it, so you were then able to use it, so you were then able to understand it.