- Math skills to hone -
(Please ignore my typos!)

1. the definitions, which are in bold. (Trivial solution, consistent system, and homogeneous system will be used a lot in this course.)
2. Examples 3, 4, and 5
3. Blue Box on Page 6
4. Last paragraph of the section on page 8

1. What are the differences between the linear systems in Examples 3, 4, and 5 in terms of the set of solutions? Are there any similarities?
2. Prove (or at least be prepared to discuss where you got stuck in proving) the following proposition:
   

   Proposition. Let $a,b,c,d$ be arbitrary real numbers. If $ad-bc \neq 0$, then the homogeneous system \[ \begin{array}{rcrcr} ax & + &by & = & 0\\ cx & + & dy & = & 0 \end{array} \] has only the trivial solution.

- Solving problems! -
(each picture has two different approaches)

1. the definitions, paying special attention to $\mathbb{R}^n$, linear combination, and traspose
2. notation for matrices, e.g. $A = [a_{ij}]$
3. summation notation

1. When we write "let $A=[a_{ij}]$ be an $m\times n$ matrix," what is the difference between $[a_{ij}]$ and $a_{ij}$?
2. True or False: every vector in $\mathbb{R}^3$ can be written as a linear combination of the following three vectors: \[\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\1\\0\end{bmatrix}, \begin{bmatrix}0\\0\\1\end{bmatrix}\]
3. True or False: every vector in $\mathbb{R}^3$ can be written as a linear combination of the following three vectors: \[\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\2\\0\end{bmatrix}, \begin{bmatrix}3\\4\\0\end{bmatrix}\]

1. the definitions, paying special attention to the definition of matrix multiplication (Def. 1.7)
2. Example 9
3. the discussion about a matrix-vector product (between examples 11 and 12)
4. Examples 11, 12

1. True or False: if $A$ and $B$ are any (arbitrary) $3\times 3$ matrices, then $AB=BA$.
2. True or False: if $A$ and $B$ are any $2\times 2$ matrices such that $AB = 0_{2\times 2}$, then it must be that either $A = 0_{2\times 2}$ or $B = 0_{2\times 2}$. Note: here $0_{2\times 2}$ denotes the $2\times 2$ zero matrix, i.e. \[0_{2\times 2} = \begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}.\]
3. True or False: if $A$ is any $2\times 2$ matrix and $c\in \mathbb{R}$ is any scalar such that $cA = 0_{2\times 2}$, then it must be that either $A = 0_{2\times 2}$ or $c=0$.

1. the theorems, which tell us that the matrix operations satisfy some familiar properties, e.g. commutativity of addition and associativity of addition and multiplication
2. Examples 9 and 10 (and the blue boxes surrounding them), which tells us that some familiar properties are not true (in general)

1. Bring in your (typed up, if possible) solution to #28(b) from 1.3; make sure that it is a formal write-up with all variables properly defined, correct punctuation, etc. Be prepared to talk through it on the document camera.

1. the definitions, paying special attention to invertible (nonsingular) and inverse
2. Examples 11 and 12

1. Give an example of a $2\times 2$ matrix that is not the zero matrix and is not invertible.
2. If $A$ is an $n\times n$ diagonal matrix such that every entry on the main diagonal is nonzero, show that $A$ is invertible. Hint: to do this, you just have to write down what the inverse is and then check that it works. If you have trouble seeing it, try experimenting in the $2\times 2$ world first.

1. Theorem 1.6
2. Examples 14

1. Take another crack at this one: if $A$ is an $n\times n$ diagonal matrix such that every entry on the main diagonal is nonzero, show that $A$ is invertible. Hint: take an educated guess at what the inverse is (based on class today), and call it $B$. To show that $B$ is indeed the inverse, you just need to show that $AB = I$ and $BA = I$ (the remark between Definition 1.10 and Example 10 simplifies this further).
2. Let $A = \begin{bmatrix} 2 & 3\\ 5 & 7\end{bmatrix}$. First, show that $B = \begin{bmatrix} -7 & 3\\ 5 & -2\end{bmatrix}$ is the inverse of $A$. Hint: you just need to show that $AB = I$. Second, use that $B = A^{-1}$ to quickly solve $\begin{bmatrix} 2 & 3\\ 5 & 7\end{bmatrix} \begin{bmatrix} x_1 \\ x_2\end{bmatrix} = \begin{bmatrix} 1\\ 1\end{bmatrix}$ (as in Example 14).

Theorem (Jon-Jenny-Atticus). If $A =[a_{ij}]$ is an $n\times n$ diagonal matrix such that every entry on the main diagonal is nonzero, then $A$ is invertible and \[A^{-1} = \begin{bmatrix} \frac{1}{a_{11}} & 0 &\cdots & 0\\ 0 & \frac{1}{a_{22}} & \cdots & 0\\ \vdots & & \ddots & \vdots\\ 0 & 0 & \cdots & \frac{1}{a_{nn}}\end{bmatrix}.\]

1. the discussion on the top half of page 57 (pay attention to the definitions)
2. Examples 2 and 3

1. Bring in your typed up solution to AP #2; make sure that it is a formal write-up that is fully justified with correct punctuation.

1. the definitions, e.g. statement, negation, conjunction, disjunction, implication, logically equivalent
2. the examples
3. De Morgan's Laws, Law of Double Negation, Law of Implication

- No Josh! -

1. the definitions
2. Examples 1.18, 2.2, 2.4

1. the definitions
2. Examples 3.1, 3.2, 3.3, 3.4

1. Start by reading Theorem 2.3 on page 96. It contains terms that we haven't defined yet, but it motivates Section 2.1.
2. Look back, if needed, to remember how we change between linear systems and augmented matrices (see pg 27-28).
3. Read Definitions 2.2 and 2.3 in Section 2.1, and then read Examples 3 and 4 in Section 2.1.

1. Do you see similarities between the manipulations we do to solve linear systems (see blue box on page 6) and the elementary row operations? Differences?
2. Consider the following linear system: \[\begin{align*}x_2 + x_3 &= 2\\x_1 + x_3 &= 1\\-3x_1 + 2x_2 &= 4\end{align*}\]
   • Write the augmented matrix corresponding to this linear system.
   • Show that the augmented matrix you found is row equivalent to \[ \left[\begin{array}{ccc|c}1 & 0 & 1& 1\\ 0 & 1 & 1& 2\\0 & 0 & 1& 3\end{array}\right].\] That is, use elementary row operations to transform the augmented matrix you found before into this one.
   • Write out the linear system corresponding to this new augmented matrix.

1. Examples 3, 6, and 10

1. Notice that, as in Example 6, you can stop the reduction process at an REF and solve the system with "back substitution", or you can continue the reduction process to RREF and have the solutions "right there." Which one do you like better? Does one method seem more efficient?
2. Suppose you are trying to solve a homogeneous linear system of 3 equations in 4 unknowns. Explain (and don't just quote a theorem) why the system must have infinitely many solutions (regardless of what the system actually looks like).
3. Is an arbitrary linear system of 3 equations in 4 unknowns guaranteed to have infinitely many solutions? Why or why not?

1. We want to show that if $A\mathbf{x} = \mathbf{0}$ has only the trivial solution, then $A$ is row equivalent to the identity. Suppose $A\mathbf{x} = \mathbf{0}$ has only the trivial solution. Using the language of pivots and free variables, find the RREF of $\left[\begin{array}{c|c}A & \mathbf{0}\end{array}\right]$. (If needed, experiment in the $3\times 3$ case first.) What does this mean about the RREF of $A$?
2. Is the converse true? That is, is it true that if $A$ is row equivalent to the identity, then $A\mathbf{x} = \mathbf{0}$ has only the trivial solution?

1. pg. 121
2. Examples 5 and 6

1. Complete the following sentence: $A$ is invertible if and only if... (insert some condition on $a$, $b$, and $c$)
   Hint: Think about what you read; there is an algorithm for finding $A^{-1}$. When does the algorithm succeed for this particular $A$ (or, alternatively, what could make it fail)?
2. What if $A$ had been an arbitrary $n\times n$ upper triangular matrix? Complete the following sentence for the $n\times n$ case: $A$ is invertible if and only if... (insert some condition on the entries of $A$)

1. Intro. through Example 5. (For the record, I find Example 1 confusing, which motivated the discussion questions below.)

1. How do you visualize a permutation? If you're at a loss, try googling "permutation."
2. How many permutations are there of the set $\{1,2,3,4\}$, i.e. how many ways are there to arrange the numbers $1,2,3,4$? (Yes, the answer is in the book. But, I want each of you to understand it and be able to clearly explain it.)
3. Is the permutation 4321 even or odd? Be able to identify the inversions.

Theorem (Mike-Ryan). If $A$ is an $n\times n$ upper or lower triangular matrix, then $A$ is invertible if and only if every entry on the main diagonal of $A$ is nonzero.

1. Read the statements of the Theorems, but you can skip the proofs for now.

1. Please construct one question of the form "provide a specific example which satisfies the given description, or if no such example exists, briefly explain why not" and add it to our review sheet here

   https://www.sharelatex.com/project/56ce03263a091c210d42ed9a

   The review sheet already contains a question I made that you can use as an example. Make sure to
   • put your initials next to your questions, and
   • put an answer in the "Answers" section of the document (again, I provided an example).
2. Repeat with a T/F question.
3. Make sure to write down your questions and answers on paper too, so you are prepared to present them in class.

Exam1ReviewQuestions.pdf

1. Example 2 in Section 3.3.

1. Let $A\in M_{n\times n}$, and assume that $A$ is invertible with $A=A^{-1}$. What are the possible values for $\operatorname{det}(A)$?

1. Examples 1, 2, 5, 6, 11 (and the blue box following Example 11)
2. The proof of Theorem 4.2(a)

1. Let $V$ be the set of all $2\times 2$ matrices (with entries from $\mathbb{R}$) that have determinant $0$. Define $\oplus$ to be ordinary matrix addition, and $\odot$ to be ordinary scalar multiplication. Is $V$ closed under $\oplus$ (see Definition 4.4(a))? That is, is it true that for all $A,B\in V$, $A+B \in V$? Is $V$ closed under $\odot$ (see Definition 4.4(b))?
2. Digest the proof of Theorem 4.2(a). Use a similar idea to prove Theorem 4.2(b).

1. Examples 7, 9, 10

1. Let $V$ be the set of all $2\times 2$ matrices (with entries from $\mathbb{R}$) that have determinant $0$. Define $\oplus$ to be ordinary matrix addition, and $\odot$ to be ordinary scalar multiplication. Is $V$ closed under $\oplus$ (see Definition 4.4(a))? That is, is it true that for all $A,B\in V$, $A+B \in V$? Is $V$ closed under $\odot$ (see Definition 4.4(b))?
2. Digest the proof of Theorem 4.2(a). Use a similar idea to prove Theorem 4.2(b).

1. Definition 4.5 through Theorem 4.3 (put your focus on Theorem 4.3)
2. Examples 1-4

1. We learned that $M_{2\times 2}$ is a vector space. Let $W$ be the subset of $M_{2\times 2}$ consisting of the $2\times 2$ matrices whose trace is $0$, i.e. $W=\{A \in M_{2\times 2} \,|\, \operatorname{tr}(A) = 0\}$. Use Theorem 4.3 to show that $W$ is a subspace of $M_{2\times 2}$.

1. Examples 6-9 and Definition 4.6

1. Be able to write on the board, from memory, the definition of a linear combination of vectors $v_1,\ldots,v_k$. I may call on multiple people for the same definition!
2. After reading Example 7 and looking over Figure 4.22, describe algebraically and geometrically the set of all linear combinations of the following two vectors in $\mathbb{R}^3$: \[\mathbf{v_1} = \begin{bmatrix}\pi\\0\\0 \end{bmatrix}\text{ and } \mathbf{v_2} = \begin{bmatrix}0\\0\\e \end{bmatrix}.\]

1. Definition 4.7, Definition 4.8, and the remark following Definition 4.8
2. Examples 1, 2, 5, 7, 8

1. Consider the following four vectors in $P_3$ \[\mathbf{v}_1 = 1+2t+3t^3,\quad \mathbf{v}_2 = 3+t+t^2-t^3,\quad \mathbf{v}_3 = -5t-t^2-t^3,\quad \mathbf{v} = -2-14t-2t^2-8t^3.\] Suppose you want to determine if $\mathbf{v}$ is in $\operatorname{span}\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$.
   • Explain why this is the same as determining if the following system is consistent. \[\left[\begin{array}{rrr|r}1 & 3 & 0 & -2\\ 2 & 1 & -5 & -14\\0 & 1 & -1 & -2\\ 3 & -1 & -1 & -8\end{array}\right]\]
   • Is $\mathbf{v}\in\operatorname{span}\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$?
2. Consider the following four vectors in $M_{2 \times 2}$ \[\mathbf{v}_1 = \begin{bmatrix}0 & 2\\ 3 & 0 \end{bmatrix},\quad \mathbf{v}_2 = \begin{bmatrix}1 & -1\\ 3 & -1 \end{bmatrix},\quad \mathbf{v}_3 = \begin{bmatrix}\pi & 0\\ 0 & -\pi \end{bmatrix},\quad \mathbf{v} = \begin{bmatrix}1 & 2\\ 3 & 4 \end{bmatrix}.\] Suppose you want to determine if $\mathbf{v}$ is in $\operatorname{span}\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$.
   • Explain why this is the same as determining if the following system is consistent. \[\left[\begin{array}{rrr|r}0 & 1 & \pi & 1\\ 2 & -1 & 0 & 2\\3 & 3 & 0 & 3\\ 0 & -1 & -\pi & 4\end{array}\right]\]
   • Is $\mathbf{v}\in\operatorname{span}\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$?

1. Definition 4.9
2. Examples 2, 6

1. Is $\{\mathbf{u}, \mathbf{0}\}$ linearly dependent?
2. Find three different choices for $\mathbf{v}\in \mathbb{R}^3$ for which $\{\mathbf{u}, \mathbf{v}\}$ is linearly dependent.
3. Can you find all vectors $\mathbf{v}\in \mathbb{R}^3$ for which $\{\mathbf{u}, \mathbf{v}\}$ is linearly dependent?

1. Theorem 4.7 (compare this with what you proved on the last writing assignment, see Remark 2 following Example 10)
2. Remarks 1-3 following Example 10 (also compare Remark 3 with what you proved on the last writing assignment)

1. Be able to write on the board, from memory, the definition of linearly independent vectors $v_1,\ldots,v_k$. I may call on multiple people for the same definition!
2. Let $\mathbf{u} = \begin{bmatrix} 1\\ 2\\ \pi \end{bmatrix}$.
   • Is $\{\mathbf{u}, \mathbf{0}\}$ linearly dependent?
   • Find three different choices for $\mathbf{v}\in \mathbb{R}^3$ for which $\{\mathbf{u}, \mathbf{v}\}$ is linearly dependent.
   • Can you find all vectors $\mathbf{v}\in \mathbb{R}^3$ for which $\{\mathbf{u}, \mathbf{v}\}$ is linearly dependent?

1. Definition 4.10
2. Example 1

1. Is $\left\{ \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}\right\}$ a basis for $\mathbb{R}^3$? Why or why not?


2. Is $\left\{ \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}\right\}$ a basis for $\mathbb{R}^3$? Why or why not?


3. Is $\left\{ \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}\right\}$ a basis for $\mathbb{R}^3$? Why or why not?

1. Definition 4.10
2. Examples 2 and 4

1. Explain, in detail, why determining if $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}$ is linearly independent is the same as determining if the following system has only the trivial solution: \[ \left[\begin{array}{cccc|c} a_1 & a_2 & a_3 & a_4 & 0\\ b_1 & b_2 & b_3 & b_4 & 0\\ c_1 & c_2 & c_3 & c_4 & 0\\ \end{array}\right] \]
2. By considering the RREF (i.e. think about pivots and free variables), explain why $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}$ is linearly dependent (no matter what $\mathbf{v}_1$, $\mathbf{v}_2$, $\mathbf{v}_3$, $\mathbf{v}_4$ actually are).
3. Explain, in detail, why determining if $\{\mathbf{v}_1, \mathbf{v}_2\}$ spans $P_2$ is the same as determining if the following system is consistent for all choices of $e,f,g\in \mathbb{R}$: \[ \left[\begin{array}{cc|c} a_1 & a_2 & e\\ b_1 & b_2 & f\\ c_1 & c_2 & g\\ \end{array}\right] \]
4. By considering the RREF, explain why $\{\mathbf{v}_1, \mathbf{v}_2\}$ does not span $P_2$ (no matter what $\mathbf{v}_1$ and $\mathbf{v}_2$ actually are).
5. Explain the following statement: "every basis for $P_2$ has the same number of vectors and that number is $\underline{\phantom{XXXX}}$".

1. Blue box on page 235
2. Example 6

1. Let $v_1 = \begin{bmatrix} 1\\ -1\\ 1\\ 0 \end{bmatrix}$, $v_2 = \begin{bmatrix} 2\\ -1\\ -1\\-1 \end{bmatrix}$, $v_3 = \begin{bmatrix} 3\\ -6\\ 12\\ 3 \end{bmatrix}$, and $v_4 = \begin{bmatrix} 4\\ -3\\ 2\\-\frac{1}{2} \end{bmatrix}$. Follow the process in the blue box on page 235 to find a basis for $W = \operatorname{span}\{v_1, v_2, v_3, v_4\}$.

1. Introduction
2. Example 1 and 2

1. Let $B_1$ be the following ordered basis for $P_2$: $B_1 = \{t^2,t,1\}$.
   
   • Explain why $\left[3t^2 + 2\right]_{B_1} = \begin{bmatrix} 3\\ 0\\ 2 \end{bmatrix}$.
   
   I'll answer this one, but make sure you understand my reasoning...

   Define $\mathbf{b}_1 = t^2$, $\mathbf{b}_2 = t$, and $\mathbf{b}_3 = 1$; thus, $B_1 = \{\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3\}$. Now, observe that $3t^2 + 2 = 3\cdot \mathbf{b}_1 + 0\cdot\mathbf{b}_2 + 2\cdot \mathbf{b}_3$, and this implies that $\left[3t^2 + 2\right]_{B_1} = \begin{bmatrix} 3\\ 0\\ 2 \end{bmatrix}$.
   

2. Let $B_2$ be the following ordered basis for $P_2$: $B_2 = \{t,1,t^2\}$.
   
   • Explain why $\left[3t^2 + 2\right]_{B_2} = \begin{bmatrix} 0\\ 2\\ 3 \end{bmatrix}$.
   
   
3. Let $B_3$ be the following ordered basis for $P_2$: $B_3 = \{t^2+t+1,t^2+t,t^2\}$.
   
   • Explain why $\left[3t^2 + 2\right]_{B_3} = \left[\begin{array}{r} 2\\ -2\\ 3 \end{array}\right]$.

1. Introduction to transition matrices (pages 261-262)
2. Example 4

1. If $A$ is invertible, show that $f$ is an isomorphism. (Remember there are 4 things to show.)
2. If $A$ is not invertible, show that $f$ is not an isomorphism. (Note, $f$ is not an isomorphism as soon as one of the four properties fails.)

1. Intro through Example 1

1. Please construct one question of the form "provide a specific example which satisfies the given description, or if no such example exists, briefly explain why not" and add it to our review sheet here

   https://www.sharelatex.com/project/570e88cfe18718921bb52d30

   The review sheet already contains some questions that you can use as an example. Make sure to
   • put your initials next to your questions, and
   • put an answer in the "Answers" section of the document.
2. Repeat with a T/F question.
3. Make sure to write down your questions and answers on paper too, so you are prepared to present them in class.

Exam2ReviewQuestions.pdf

1. Give an example of a $3 \times 4$ matrix for which the row space has dimension 2. (You can make your example as simple as you like.)
2. What is the dimension of the column space of your example?
3. Is it possible to find a $3 \times 4$ matrix for which the row space has dimension 4?
4. Is it possible to find a $3 \times 4$ matrix for which the column space has dimension 4?

1. Definition 6.1
2. Examples 1, 4
3. Theorem 6.2 as well as the paragraphs immediately before and immediately after it.

1. Definitions 6.2, 6.3, 6.4
2. Examples 2, 3, 5
3. Examples 4, 6

1. Is $L$ one-to-one?
2. What polynomials (from $P_3$) are in $\operatorname{ker}(L)$?
3. Is $L$ onto?
4. What polynomials (from $P_2$) are in $\operatorname{range}(L)$?

1. Theorem 6.6 (this should remind you of the Rank-Nullity Theorem)
2. Examples 9 and 10

1. Explain why $L$ is a linear transformation. (Make use of the properties of trace.)
2. Describe $\operatorname{range}(L)$. Is $L$ onto? What is $\dim(\operatorname{range}(L))$?
3. Describe $\operatorname{ker}(L)$. (Hint: we've talked about the matrices in $\operatorname{ker}(L)$ many times before.) Is $L$ one-to-one? Use Theorem 6.6 to quickly find $\dim(\operatorname{ker}(L))$.

1. Introduction
2. Example 1

1. We have seen that "taking the derivative" is a linear transformation. That is, for every positive $n$, the function $L:P_n\rightarrow P_n$ defined by $L(p(t)) = p'(t)$ is a linear transformation. Section 6.2 is about finding a way to view $L$ as a matrix transformation.
   • Notice that I wrote $L:P_n\rightarrow P_n$ instead of $L:P_n\rightarrow P_{n-1}$. Think about how that choice affects the following questions. What else might that choice affect, e.g. onto, one-to-one,...?
   • Let $p(t) =a + bt + ct^2$. Compare $p'(t)$ with $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0 \end{bmatrix}\begin{bmatrix}a\\b\\c \end{bmatrix}$.
   
   
   • Let $p(t) =a + bt + ct^2 + dt^3$. Compare $p'(t)$ with $\begin{bmatrix}0 & 1 & 0 & 0\\0 & 0 & 2 & 0\\0 & 0 & 0 & 3\\0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}a\\b\\c\\d \end{bmatrix}$.
   
   
   • Can you generalize the above situation? What matrix would you use to compute the derivative of $p(t)= a_0 + a_1t + a_2t^2 + \cdots a_nt^n$? Why?

1. Intro through Definition 7.1

1. Let $L:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be defined by $L\left(\begin{bmatrix}x\\y\end{bmatrix}\right) = \begin{bmatrix}y\\x\end{bmatrix}$


• Find all $\mathbf{v} \in \mathbb{R}^2$ such that $L(\mathbf{v}) = \mathbf{v}$. Can you also explain this geometrically (by thinking about $L$ geometrically)?
• Find all $\mathbf{v} \in \mathbb{R}^2$ such that $L(\mathbf{v}) = -\mathbf{v}$. Can you explain this geometrically as well?
• Find all $\mathbf{v} \in \mathbb{R}^2$ such that $L(\mathbf{v}) = 2\mathbf{v}$. Can you explain this geometrically?

1. The definitions of eigenvalue and eigenvector of a matrix in the middle of page 442
2. Definition 7.2 (this is really important!)
3. Example 11 and 12

1. Find the characteristic polynomial of $\displaystyle A=\begin{bmatrix}2 & 7\\7 &2\end{bmatrix}$. Please write the polynomial in factored form.


2. Find the characteristic polynomial of $\displaystyle A=\begin{bmatrix}1 & 0 & 0\\3 & 2 & 0\\ 0 & 0 & 1\end{bmatrix}$. Please write the polynomial in factored form.


3. If $a_{ij}$ are arbitrary real numbers, find the characteristic polynomial of $\displaystyle A=\begin{bmatrix}a_{11} & 0 & 0\\a_{21} & a_{22} & 0\\ a_{31} & a_{32} & a_{33}\end{bmatrix}$. Please write the polynomial in factored form.


4. If $A = [a_{ij}]$ is an arbitrary $n\times n$ lower triangular matrix, find the characteristic polynomial, in factored form, of $A$.

1. The definition of similar matrices on page 410

1. "Similar matrices are similar." Let $A,B \in M_{n\times n}$, and assume $B$ is similar to $A$, i.e. $B = P^{-1}AP$ for some invertible matrix $P$.
• Explain why $\operatorname{Tr}(A) = \operatorname{Tr}(B)$? Hint: use properties of the trace.
• Explain why $\det(A) = \det(B)$? Hint: use properties of the determinant.
• Can you think of other things that may be the same for similar matrices?
• Explain why $B^k = P^{-1}A^kP$ for any positive integer $k$. Hint: to see what's going on, start with $k=2$, and try to simplify $B^2 = (P^{-1}AP)^2$

1. Theorem 7.4, and the remark following Theorem 7.4
2. The discussion between Examples 3 and 4 (this, together with Theorem 7.4, is what I presented as the Diagonalization Theorem in class)
3. Theorem 7.5
4. Examples 2-6

1. Find a basis for $E_1(A)$, i.e. the eigenspace of A associated to $\lambda=1$.
2. Find a basis for $E_2(A)$, i.e. the eigenspace of A associated to $\lambda=2$.
3. Show that when your basis for $E_1(A)$ is combined with your basis for $E_2(A)$, you get a basis for $\mathbb{R}^3$, and explain why this shows that $A$ is diagonalizable. (Hint: to see why $A$ is diagonalizable, use Theorem 7.4, i.e. the Diagonalization Theorem.)
4. Find a basis for $E_1(B)$ and a basis for $E_2(B)$.
5. Explain why $B$ is NOT diagonalizable. (Hint: use the Diagonalization Theorem; remember it is an if and only if statement.)

https://www.sharelatex.com/project/572cecd7fddf965455c829ca