### Test 2 Study Guide

This test will be closed to notes/books, but a calculator will be allowed
(but no cell phone nor other calculators bundled in combination with
additional technologies).
There will be various components
to the test and your grade will be based on the
quality of your responses in a timed environment
(turned in by the end of class).
I suggest that you review your class notes and go over ASULearn
solutions [since ASU is down I have listed some of the recent solutions here:]
to the practice problems and problem sets.
Prob Set 4
Prob Set 5
Practice 4.4-4.6

Here are the topics we have been focusing on:
test 1 material (be sure to study test 1 and any related material that
you feel that you need to brush up on, including the derivations there)
algebra and geometry of vectors
algebra and geometry of linear combinations
algebra and geometry of row operations and determinants
vector spaces
linearly independence
span
basis
dimension
algebra and geometry of columns
algebra, geometry and basis of homogeneous solution spaces
writing out the solutions of a system as a homogeneous plus particular
portion.
mixing problems

**Some Maple Commands**
Here are some Maple commands you should be pretty familiar with by now
for this test - i.e. I will at times show a command, and it may be
with or without
its output:
**
**

> with(LinearAlgebra): with(plots):

> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);

> ReducedRowEchelonForm(A);

> GaussianElimination(A); (only for augmented
matrices with unknown variables like
k or a, b, c in the augmented matrix)**
**

> B:=MatrixInverse(A);

> A.B;

> A+B;

> B-A;

> 3*A;

> A^3;

> evalf(A^100); or** evalf((A^100).U); **(be
careful to use fractions for stochastic matrices)**
**

> Determinant(A);

> Vector([1,2,3]);

> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2); plot vectors as line segments in R^{3}
(columns of matrices) to show whether the the columns are in the same plane,
etc.
**
**

> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);

> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4);
plot equations of planes in R^3 (rows of augmented matrices) to look
at the geometry of the intersection of the rows (ie 3 planes intersect in
a point, a line, a plane, or no common points)

**Fill in the blank**

There will be some fill in the blank short answer questions, such as
providing:
definitions related to any of the above topics, including test 1 material
real-life applications, like ____ is a real-life application of matrix
inversion (where the natural answer would be the Hill cipher)
fill in the blank related to computations, examples and
interpretations

**Calculations and Interpretations**

There will be some by-hand computations and interpretations,
like those you have had previously for homework, clicker questions
and in the problem sets. For instance,
review the cement problems from ps 4 and 5
carefully, in addition to the practice problems,
like the Maple practice problem.

**Short Derivations**

There will be some
short derivations - the same as or similar to what we have seen before, like:
Those on test 1
Why things are or are not vector spaces and subspaces, and know the
reasons and relevant counterexamples of axioms 1 and 6
(the addition and scalar
multiplication axioms).
If n is given as the number of rows or columns, but you are asked to leave n
general, be sure to do so (ie do not define n to be 2 or 3 - instead, use
the general nxn identity matrix, or the nxn matrix with all 0s except a 1 in
the top left corner, or something similar for your examples), like in ps 4
solutions.
The proof that in a basis, each vector can be written as a unique
linear combination of the basis vectors.
The proof that the only subspaces of R^{2} are
the 0 vector, lines through the origin, and all of R^{2}

**True/False**
True/False statements and counterexamples has been a recurring theme
in all of the chapters, so you can expect problems like we have seen
before in practice problems, problem sets (solutions on ASULearn),
and the i-clickers (questions on the class highlights page).
Be sure that you know
how to find counterexamples
There will be questions were you answer true or false and if
false, then you will either correct the text after a given word like
"then" or provide a counterexample.

For example,
It is possible to find
a set of 4 vectors that span R^{3} but are not linearly independent
[make your life easy and choose a simple counterexample, like
the standard R^{3} basis
(1,0,0), (0,1,0), (0,0,1) which already spans R^{3}
but add in the (0,0,0) vector or any other vector to get rid of
efficiency)],
It is not possible to find a set of 3 vectors that span R^{3}
but are
not linearly independent [a theorem says if you have the
correct number of elements for a basis in a known vector space with a
given dimension, then span is true iff linearly
independence is]
It is not possible to find a set of 2 vectors that
span R^{3} [one needs 3 vectors to
represent R^{3} - 2 non-trivial
vectors can at most span a plane (if they are
linearly independent) or a line (if they are linearly dependent multiples
of each other)].
It is possible to find a set of 2 vectors that are linearly independent
in R^{3}
but do not span R^{3}, by taking 2 vectors that span a plane in
R^{3}.
If a vector w is a linear combination of vectors in V, then adding w to those vectors will force the set to be not li (w=c1v1+... means that 0 = -w+ c1v1+... so there is a nontrivial solution to the homogeneous equation). On the other hand we have seen examples where w is not a linear combination of vectors in a set, but the set is not li, because one of the other vectors is a linear
combination.
You should also be able to answer questions like this for
R^{2} also! For example - you can find 1 linearly independent vector
in R^{2} that does not span R^{2} -
(1,2) or any non-zero vector will do, which spans a line with the given
slope 2=(y-0)/(x-0). However 2 li vectors in
R^{2} would not be possible, since two vectors in R^{2}
that are linearly independent will also span.
an example of a set that spans R^{2} but is not linearly
independent,
an example of 2 different basis sets for R^{2},
an example of a set that is li in R^{2}
but does not span,
subspaces of R^{2} [0 vector, lines through origin,
R^{2}],and R^{3}
[0 vector, lines through the origin, planes through the origin,
R^{3}],...
examples that show matrix multiplication is or is not commutative
(like rotation matrices that are commutative),
examples of systems that have 0, 1 or infinitely many solutions to them,
algebra and geometry of objects like Gaussian reductions like
t (row 1) + (row 2) [parallel to row 1 through the tip of row 2 and
the operation preserves area or volume which is the determinant]
or linear combinations like c(column 1) +d (column 2) [a plane
through the origin if the columns
are l.i. and a line through the origin otherwise].
and more: others like we have seen in homework, clicker
questions, problem sets...