Spring 2017

This sheet specifies a selection of exercises from the text, and in
some
cases a few additional exercises, for the first several weeks of class.
The
notation 1-33o means the odd problems between 1 and 33,
inclusive. Please follow these format instructions
. For problems that require
you to find the rref of a matrix, once you have mastered the row
operation process, feel free to use software such as
freemat. Note: you can find rref at wolframalpha.com by entering
an input of the form rref([1,2,3],[4,5,6]).

- Give correct definitions of the following terms or expressions:
*linear system, solution set, equivalent, inconsistent, row equivalent*. - Give an example of a system of 2 linear equations in 2 unknowns with no solution.
- (*) Can a system of linear equations have just one equation?
- (*) Is the following a linear system?
- Is it possible for a linear system of two equation in two unknowns to have exactly 1 solution? Exactly 2 solutions? Exactly 100 solutions? An infinite number of solutions? For each question, if you say it is not possible, explain why. If you say it is possible, give an example.

x^{2} + y^{2} |
= | 13 | |

x^{2} -y^{2} |
= | 5 |

Explain why someone might say that the system is linear in *x*^{2}
and *y*^{2}. Show how methods for linear
systems can be used to solve this system of equations for x and y.

- Two students are doing linear algebra homework. They each start with the same matrix and use row operations to obtain echelon form. When they compare their work, they see that they did not get the same answer. Did someone make an error?
- One of the two students of the previous problem suggests,
*Let's keep going until we get to reduced echelon form. Then if one of us made an error we will know for sure because we will still have two different answers.*Do you agree with this reasoning? - (*) The book defines pivot
*position*and pivot*column*, but does not define pivot*row.*Which rows in the rref of a matrix are pivot rows? Non-pivot rows? - How can you tell by looking at the reduced echelon form of the augmented matrix of a linear system whether or not there are any free variables? How can you tell which variables are the free variables and which are the basic variables?
- (*) If a system has more variables than equations, is it possible for it to have a unique solution?

- (*) Can be expressed as a linear combination of and ?
- Can be expressed as a linear combination of and ?
- Define
*linear combination.*Explain what is meant by the*set of vectors spanned by another set of vectors.* - In the blue box on page 34, the vector equation

- (*) Suppose that vectors
**u**,**v**, and**w**are given, and you notice that**w**=**u**+**v**. Explain why every vector in Span {**u**,**v**,**w**} is also in Span{**u**,**v**}. - (*) If it is true that every vector in Span {
**u**,**v**,**w**} is also in Span{**u**,**v**}, must**w**be somehow made up from**u**and**v**, as in the last problem? Explain. (Note, do not assume that the vectors in this problem are the same ones described in the preceding problem.)

x_{1} a_{1} + x_{2}a_{2}
+ ... + x_{n}a
= _{n}b |

appears. In this equation, which letters are the variables? Which
stand for constants? Give a specific example of such an equation with *n*
=
4, and with the variables left as letters, but with the constants
replaced with some specific values.

- At the middle of page 43 there is an example in which an
augmented
matrix has a final column given by
. This final column can be written as a product
*C***b**where*C*is a 3 by 3 matrix, and**b**is the vector . Find the matrix*C*. - Proving properties of matrix operations: In general, to prove
properties like those shown in theorem 5, you must prove that two
things are equal, namely,
the final result on either side of the equal sign. When the two items
are
vectors, the usual way to do this is to develop a formula for the entry
in
position
*j*(called the*j-th*entry) of each side, and show that they are equal. Do this for the two equations in theorem 5, using the following fact: if the*ij*entry of the*m ×**n*matrix*A*is*a*and if the_{ij}*j*^{th}entry of vector**v**is*v*then for any_{j}*i*, the*i*^{th}entry of*A***v**is given by .

- Define homogeneous system, trivial solution, nontrivial solution, parametric vector equation
- Three students are independently working on solving an
inhomogeneous
linear system of the form
*A***x**=**b**. The first student finds a solution vector**x**. The second student finds a different solution vector**y**. Did someone make an error? A third student finds yet another solution,**z**. The three students notice that**x**+**y**=**z**. Did someone make an error? - (*) Prove: If
**a**and**b**are solutions to a homogeneous system, then so is**a**+**b**. - Prove: If
**a**is a solution to a homogeneous system, then so is*r***a**for any real number*r*. - (*) Prove: If
**a**and**b**are solutions to a homogeneous system, then so is*r***a**+*s***b**, for any real numbers*r*and*s*.

- Try to complete the following statements of theorems and/or definitions without looking them up in the book:
- A set of
*two*vectors is linearly dependent if and only if ... - If a set of vectors in
**R**contains the zero vector then ...^{n} - For a set of vectors, if the number of vectors in the set is more than the number of entries in each vector, then the set is ...
- The columns of a matrix
*A*are linearly independent if and only if the matrix equation*A***x**= 0 ... - If
*S*= {**v**_{1}, ...,**v**} is a set of two or more vectors, then_{p}*S*is a linearly dependent set if and only if ... - (*) Prove: If {
**v**_{1},**v**_{2}, ...,**v**} is a dependent set of vectors, then so is {_{n}**v**_{1},**v**_{2}, ...,**v**,_{n}**v**_{n}_{+1}} for any vector**v**_{n}_{+1}. - (*) Prove: If {
**v**_{1},**v**_{2}, ...,**v**} is an independent set of vectors, then so is any non-empty subset. To simplify the argument, you may assume that any non-empty subset can be expressed as {_{n}**v**_{1},**v**_{2}, ...,**v**} for some_{k}*k*with 1__<__*k*__<__*n*. Why is that valid?

- Define the following terms: diagonal entries, diagonal matrix, row-column rule, commute, transpose.
- Suppose
*A*and*B*are 2 × 2 matrices. For each of the following equations, write*Always*if the equation is true for all*A*and*B*,*Sometimes*if the equation is true for some*A*and*B*but not for others, and*Never*if the equation is not true for any*A*and*B*. For each equation, give an explanation of your answer. - (
*A*+*B*)^{2}=*A*^{2}+2*AB*+*B*^{2} - (
*A*+*B*)(*A*-*B*) =*A*^{2}-*B*^{2} - (
*A*^{3})^{5}=*A*^{15} - A
^{3}*A*^{5}=*A*^{8} - (
*AB*)^{3}=*A*^{3}*B*^{3} - (*) The blue box on page 102 indicates that the columns of
*AB*must be linear combinations of the columns of*A*. Show that the reverse is not true. That is, show (by example) that the columns of*A*need not be linear combinations of the columns of*AB*. - (*) A matrix
*A*is called*symmetric*if*A*=^{T}*A*. Show that for any matrix*A*,*A*and^{T}A*AA*are symmetric. Does^{T}*A*have to be symmetric?^{T}AA^{T} - (*) Let
*A*and*B*be*n*×*n*matrices. Suppose that the system*A***x**=**O**has a nontrivial solution. Does*BA***x**=**O**have to have a nontrivial solution? How about*AB***x**=**O**? - (*) Prove: If
*A*and*B*are*n*×*n*matrices and if the columns of*A*are linearly dependent then the columns of*BA*are linearly dependent as well.

**Section 2.2 **1-11o, 12, 13, 15, 16*, 21-24, 29, 31,
33*,35. Plus:

- Define the terms: invertible, inverse, elementary matrix.
- State a theorem that concerns the existence and uniqueness of
solutions for a system
*A***x**=**b**under the assumption that*A*is invertible. - State a theorem that concerns inverses of products and transposes of invertible matrices.
- State a theorem that concerns the row operations, row
equivalence, and invertibility of a matrix
*A*. - (*) Find the conditions under which a 2 ×
2 matrix
*A*satisfies*A*^{-1}=*A*. Prove that^{T}*A*^{-1}=*A*if and only if your conditions hold.^{T} - (*) We say that a square matrix
*A*satisfies a polynomial*p*(*x*) =*c*+_{n}x^{n}*c*_{n}_{-1}*x*^{n}^{-1}+ ...+*c*_{1}*x*+*c*_{0}if(

p*A*) =*c*+_{n}A^{n}*c*_{n}_{-1}*A*^{n}^{-1}+ ...+*c*_{1}*A*+*c*_{0}*I*= 0.

Suppose*A*satisfies the polynomial*p*(*x*) =*x*^{3}+ 4*x*^{2}- 7*x*+2. Show that*A*is invertible, and find a formula for its inverse.

**Determinant Handout: **Read the handout and do the problems
(from our text) listed at the end.