Learning the Language of Linear
Algebra
John Hannah (Canterbury, NZ)
Sepideh Stewart (Oklahoma, US)
Mike Thomas (Auckland, NZ)
Summary
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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 u1, u2, u3 be vectors in 2-space. Does u3
usually belong to the span of u1, u2?
• Are 3 vectors in 2-space usually linearly
independent?
• Interpret your results geometrically.
Sample response 1
Given three vectors u1, u2 and u3 in 2-space, u3 is
usually in the span of u1 and u2. This is because
it is impossible for u1, u2 and u3 to all be linearly
independent in 2-space as any two linearly
independent vectors would span all of 2-space,
thus u3 would be a linear combination of u1 and
u2, and thus in the span of u1 and u2 …
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 R3 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.