Test 1 Study Guide

Test 1 Study Guide: Sections from 1.1-1.5, 1.7 and applications

It is time for our first test in order to be sure that everyone reviews and understand some of the fundamental concepts. 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 test will end at the end of class - you may leave early if you are finished)

Definitions

Computations and Interpretations / Analyses

Conceptual 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." I suggest that you review your class notes and go over ASULearn solutions to the practice exercises and problem sets as well as the clicker questions. Questions will be typeset using LaTeX (like this) and will typically be formatted as follows:

Fill in the Blank

Short Answer Calculations and Interpretations / Analyses (like Problem Set 2 cement mixing with various parts to a question).

True/False Questions: You may be asked to identify true statements [no need to recall page #s nor Theorem #s like in problem sets], provide counterexamples, and/or correct statements

Here are the topics we have been focusing on:

algebra and geometry of equations and their solutions; algebra and geometry of rows of a matrix

Gaussian and Gauss-Jordan methods and history

augmented matrix, coefficient matrix

pivots/leading 1s

unique, 0 or infinite solutions algebraically and geometrically

parametrization of infinite solutions

lines or planes interesecting in a point, line or plane according to the number of free variables in a parametrization

homogeneous systems and their solutions

equations meeting certain critera and their solutions or consistency [like 2 equations in 3 unknowns, 3 equations in 2 unknowns...]

algebra and geometry of vectors; algebra and geometry of columns of a matrix

algebra and geometry of objects like Gaussian reductions like
t (row 1) + (row 2) [parallel to row 1 through the tip of row 2]
c(column 1) +d (column 2) [a plane through the origin if the columns are l.i. and a line through the origin otherwise, unless the columns were trivial 0 vector].

writing out the solutions of a system as a vector parametrization equation with homogeneous plus particular portions

diagonal of a parallelogram, scaling along a line

scalar multiplication, addition and matrix multiplication of matrices and vectors (and relationship to systems of equations)

algebra and geometry of linear combinations and weights; mixing problems

applications of linear combinations to manufacturing, physics...

do we span R², R³,...

what do we span?

is a vector in the span?

linearly independent

span but not linearly independent; linearly independent but not span

practical applications of span; of linearly independent

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 b1, b2, b3 in the augmented matrix)
> Vector([1,2,3]);
> 3*S + 5*A + 2*L;
> evalf(M) decimal approximation of 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) Some Sample Test Questions Fill in the blank

Set up the augmented matrix corresponding to the definition of span for these vectors... __________ (vectors as columns and then a column of b_i s).

________ is a real-life application of a linear combinations of vectors (center of gravity, cement mixing)

To satisfy the definition of span, we can have _______ solutions (1 or infinite)

Homogeneous means ______

Gaussian means ______

Trivial solution means ______

Unique means ______

Computations and Interpretations/Analysis:

There will be some by-hand computations and interpretations, like those you have had previously for homework, clicker questions and in the problem sets. You should be comfortable with by-hand Gaussian Elimination and multiplication of a Matrix and a vector, for example. You are not expected to remember page numbers or Theorem numbers, but you are expected to be comfortable with definitions, "big picture" ideas, computations, analyses... Be sure to review cement mixing from problem set 2, for example.

True/False

Correct after the word then or write true:
If A is an nxn matrix, and x and b are nx1 vectors, then the matrix-vector equation Ax=b has 0, 1 or infinite solutions. (True)

Correct after the word has or write true:
A consistent augmented matrix with a row of 0s has infinite solutions (False: correction is 1 or infinite solutions)

Provide a counterexample or write true:
A consistent augmented matrix with a row of 0s has infinite solutions (False: Matrix([[1,0,5],[0,1,2],[0,0,0]])

Most true/false questions will connect to things we have seen, although they may be rephrased or repackaged. For example, we have had a true false question in hw asking about whether a homogeneous linear system is always consistent. The question could be stated just like it was, but there are also lots of related questions that one could ask, such as:
-Correct after "can have" or write true: Ax=0, with A mxn, x nx1 and 0 mx1 can have 0, 1 or infinite solutions. (Correction: 1 or infinite solutions (as the trivial (0 vector solution for the weights) always works)
-Correct after "system" or write true: A homogeneous linear system can be inconsistent. [Correction: can never be inconsistent]
-Provide a counter-example or write true: A homogeneous linear system with 2 equations in 3 unknowns has exactly one solution. (False x+y+z=0, 0x+y+z=0 is missing a pivot for z so it has infinite sols)
-Write true or correct after "has": A homogeneous linear system with 2 equations in 3 unknowns has exactly one solution. (Correction: has the trivial solution and other solutions (ie infinite solutions because there will be a missing pivot for a variable)
-Write true or correct after "has": Ax=0 with the columns of A linearly independent has only the trivial solution (True)

Clicker questions and true/false from the hw and during class review, as well as other material we saw during class is a good review for this!