### Test 2 Study Guide (cumulative + 2.1-2.3, 1.8 (p. 62, 65, & 67-68), 1.9 (p. 70-75), 2.7 and applications)

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 types of questions on the test and your grade will be based on the quality of your responses in a timed environment.

The formatting will be just like test 1. The test is cumulative but will focus on new material. Here is a

• sample test questions
• solutions

As listed there and on test 1 you will see three sections that are typset formally (using LaTeX):

• Fill in the Blank
• Computations and Interpretations / Analyses
• True/False Questions
• As such the test will be a mixture of computational and definition questions as well as critical reasoning and questions involving the "big picture." Most questions are adapted from or taken right from exercises we had for homework, problem set questions, and clicker questions, although they may be rephrased or repackaged to further develop develop critical thinking and problem-solving skills (I'm trying to help you develop your linear independence!), so I suggest that you review those solutions and any related material you need to brush up on. Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.

Here are the topics we have been focusing on: (and here is the brief overview from class
• Test 1 [test 1 study guide] including definitions such as linearly independent, span... Test 2 will be a majority of new material but still have a small portion of test 1 material - questions from test 1 may appear on test 2.
• Algebra of matrix multiplication [the definitions of AB, the number of multiplications in AB]
• Elementary Matrices like Matrix([[1, 0, 0], [0, 1, 0], [-2, 0, 1]]) representing the row operation r_3' = -2r_1 + r_3 or Matrix([[1,0],(-2,0]]) representing the row operation r_3' = -2r_1 + r_2
• Inverse of a Matrix as a concept A.A-1=I and in Maple and the computational formula for the inverse of a 2x2 matrix
• Transpose of a Matrix
• Matrix algebra properties that do hold, often reasoned using some combination of applying an inverse (if it exists) to both sides of an equation, reorder parenthesis by associativity, cancel A by its inverse, and reducing the identity to get whatever is left alone. [like Ax=b has 1 unique solution x=A-1b when A-1 exists and x and b are the correct size column vectors...] OR reasoned using arguments involving pivots/missing pivots
• Matrix algebra properties that don't hold and counterexamples [like 2 non-zero matrices that multiply to yield a zero matrix, AB is not necessarily BA...]
• Square matrix theorem (last slide of What Makes You Invertible):
The following are equivalent for square matrices A_nxn: A invertible, A reduces to the Identity matrix, columns A span Rn, A has full row pivots, columns A linearly independent, A has full column pivots, Ax=b has 1 unique solution x=A-1b, the linear transformation T(x)=Ax is has range all of Rn...
• Negations of the square matrix theorem for non-invertible A_nxn matrices, like a square nxn matrix A that does not row reduce to the identity matrix is logically equivalent to the columns of A not spanning Rn.
• Examples and counterexamples of the square matrix theorem when A is not a square matrix [like examples of a matrix whose column vectors span but are not l.i...]

Applications:
• Hill Cipher: coding using A.uncoded message and decoding using A-1.coded message
• Condition Number [you do NOT need to know the formulas - just the big picture idea]
• Linear Transformations of the plane, both 2x2 as well as 3x3 homogeneous coordinates versions. Know the following (which are on the review LaTex slide):
general rotation matrix
projections onto the y=x line, and the x and y axes
reflections across the y=x line, and the x and y axes
horizontal shear
dilation
translation
• Rotate about a point, like (4,9): Translate by (4,9).Rotate.Translate by (-4,-9)
• Composition of linear transformations: right to left (ABCx means first C(x) then B applied to that then A applied to that result, just like with function composition)
• Big picture ideas of Yoda and transpose of a matrix
• Computer speed of using associativity on (AB).Large matrix=A(B.Large matrix) including counting the number of multiplications and reasoning that (AB).Large matrix is much faster
• Decimals versus fractions in a computer algebra software program like Maple
• 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)
> Transpose(A);
> ConditionNumber(A);
(only for square matrices)
> Vector([1,2,3]);
> B:=MatrixInverse(A);
> A.B;
> A+B;
> B-A;
> 3*A;
> A^3;
> 3*S + 5*A + 2*L;
> evalf(M)
> 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 R3 (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);
> display (a,b,c);
> 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)