Math 301 course resources

- Complete the activity from Lecture 22.
- Complete the activity from Lecture 23.
- Diagonalize the matrix $A=Q\Lambda Q^T$.

$\begin{bmatrix}1&0&2\\0&-1&-2\\2&-2&0\end{bmatrix}$ - Complete the activity from Lecture 24.
- Find the singular value decomposition $A=U\Sigma V^T$.

$\begin{bmatrix}1&2\\3&6\end{bmatrix}$

- Find the limit distribution of a Markov process with the given matrix. (In other words, find the eigenvector corresponding to $\lambda=1$.) $(a)\begin{bmatrix}1&.2\\0&.8\end{bmatrix} (b)\begin{bmatrix}.2&1\\.8&0\end{bmatrix} (c)\begin{bmatrix}.5&.25&.25\\.25&.5&.25\\.25&.25&.5\end{bmatrix}$
- Suppose the $2\times2$ matrix shown below is known to be a Markov matrix (the columns sum to $1$). What is the eigenvector corresponding to $\lambda=1$?

$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ - Divide a population into $15$-year groups: $0-15$ up through $75-90$. Suppose the fertility of the groups are as follows:

$F_1=0,\quad F_2=.6,\quad F_3=.4,\quad F_4=0,\quad F_5=0,\quad F_6=0$

Suppose the survival of the groups are as follows:

$S_1=1,\quad S_2=1,\quad S_3=.95,\quad S_4=.9,\quad S_5=.65$

Write the Leslie matrix modeling this population. Find the overall growth rate of the population (the dominant eigenvalue). Find the limit population distribution. (Use a computer to help with the calculations.) - For each matrix $A$, find $e^{At}$.

$(a)\begin{bmatrix}4&5\\5&4\end{bmatrix} (b)\begin{bmatrix}6&-2\\2&1\end{bmatrix}$ - The rabbit and wolf populations interact according to $r’=6r-2w$, and $w’=2r+w$. Suppose there are initially 30 rabbits and 30 wolves. Find formulas for the populations $r(t)$ and $w(t)$ of each species over time. After a long time, what is the ratio of rabbits to wolves?
- If $A$ has eigenvalues $\lambda_1$ and $\lambda_2$, what are the eigenvalues of $e^A$? Explain your answer.

- Find the eigenvalues and eigenvectors of the matrices.

(a) $\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}$ (b) $\begin{bmatrix}1&2&3\\0&4&5\\0&0&6\end{bmatrix}$ (c) $\begin{bmatrix}1&2&1\\3&6&3\\4&8&4\end{bmatrix}$ - What is the relationship between the eigenvalues/vectors of $A$ and $A^2$? Try it on the matrices:

$A=\begin{bmatrix}-1&3\\2&0\end{bmatrix},\qquad A^2=\begin{bmatrix}7&-3\\-2&6\end{bmatrix}$ - What is the relationship between the eigenvalues/vectors of $A$ and $A+I$? Try it on the matrices:

$A=\begin{bmatrix}1&4\\2&3\end{bmatrix},\qquad A+I=\begin{bmatrix}2&4\\2&4\end{bmatrix}$ - What is the relationship between the eigenvalues/vectors of $A$ and $A^{-1}$? Try it on the matrices:

$A=\begin{bmatrix}0&2\\1&1\end{bmatrix},\qquad A^{-1}=\begin{bmatrix}-1/2&1\\1/2&0\end{bmatrix}$ - For each of the following matrices, diagonalize the matrix if possible. If you can’t then explain what went wrong.
- $A=\begin{bmatrix}1&3\\0&5\end{bmatrix}$
- $B=\begin{bmatrix}3&1\\0&3\end{bmatrix}$
- $C=\begin{bmatrix}1&1\\2&2\end{bmatrix}$
- $D=\begin{bmatrix}0&0&-2\\1&2&1\\1&0&3 \end{bmatrix}$

- Use your diagonalization from problem 5(a) to calculate $A^{100}$.
- In each of the following, find an example of a $2\times 2$ matrix $A$ with the given properties, and verify your example, or else explain why it is impossible.
- $A$ has two real eigenvalues and is diagonalizable
- $A$ has two real eigenvalues and is not diagonalizable
- $A$ has two complex eigenvalues and is diagonalizable
- $A$ has one real eigenvalue and is diagonalizable
- $A$ has one real eigenvalue and is not diagonalizable

- For each of the following matrices, find the determinant using the formula $ad-bc$. Then, if possible, find the inverse of the matrix.

(a) $\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$ (b) $\begin{bmatrix}1-2\cos^2\theta&-2\cos\theta\sin\theta\\-2\cos\theta\sin\theta&1-2\sin^2\theta\end{bmatrix}$ - Use elimination and the properties of the determinant to find the determinant of each matrix.

(a) $\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$ (b) $\begin{bmatrix}t&1&t^2\\t&1&t\\t^2&t&1\end{bmatrix}$ (c) $\begin{bmatrix}a&b&x\\c&d&y\\0&0&1\end{bmatrix}$ (d) $\begin{bmatrix}a&0&b\\0&5&0\\c&0&d\end{bmatrix}$ (e) $\begin{bmatrix}3&-1&0&0\\-1&3&-1&0\\0&-1&3&-1\\0&0&-1&3\end{bmatrix}$ - Show that the value of the determinant of the following matrix is not affected by the value of $x$. Then write down your
*own*example of a matrix, with an $x$ in it, whose determinant is not affected by the value of $x$.

$\begin{bmatrix}1&1&x\\1&2&2\\1&2&5\end{bmatrix}$ - Decide whether each of the following is true or false, and in either case explain your answer.
- The determinant of $A+I$ is the determinant of $A$ plus $1$
- The determinant of $AB$ is the determinant of $A$ times the determinant of $B$
- The determinant of $4A$ is $4$ times the determinant of $A$

- Use Cramer’s formula to find the inverse of the matrices.

(a) $\begin{bmatrix}1&2&0\\0&3&0\\0&7&1\end{bmatrix}$ (b) $\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix}$ - Given an $n\times n$ matrix $A$ with determinant $\det(A)$, what is the detereminant of the cofactor matrix $C$ and why? [Hint: Use $AC^T=\det(A)I$.]
- Find the area of the paralellogram or parallelepiped determined by the given vectors.

(a) $(3,2)$, $(1,4)$ (b) $(1,-1,0)$, $(3,3,1)$, $(-1,-3,6)$ - Consider the triangle with corners $(2,1)$, $(3,4)$, and $(0,5)$. What is the area of the triangle? [Hint: a triangle is half a parallelogram.]

- Given the data set, find the coefficients of the best fit line. Then create an accurate plot of the data and the best fit line.
- $(-1,7),(1,7),(2,21)$
- $(-2,4),(-1,2),(0,-1),(1,0),(2,0)$

- In the previous question, second bullet, what was special about the data set? Use this special property to find and explain how a student can solve the problem using dot products and not matrices.
- Given the data set, find the best fit parabola. Then create an accurate plot of the data and the best fit parabola.
- $(-2,0),(-1,0),(0,1),(1,0),(2,0)$

- In each problem, let $V$ be the subspace with the given basis. Find a basis for $V^\perp$.
- Basis for $V$: $\begin{bmatrix}1\\3\\4\end{bmatrix},\quad \begin{bmatrix}5\\2\\7\end{bmatrix}$
- Basis for $V$: $\begin{bmatrix}1\\2\\2\\3\end{bmatrix},\quad \begin{bmatrix}1\\3\\3\\2\end{bmatrix}$

- Let $A$ be the given matrix. Draw the (traditional) graph of $R(A)$ and $N(A)$ on one set of axes, and the graph of $C(A)$ and $LN(A)$ on a second set of axes.

$A=\begin{bmatrix}1&2\\3&6\end{bmatrix}$ - Consider a $2\times 2$ matrix $A$ with the following properties: the sum of the rows of $A$ is the vector $(1,1)$ and the sum of the columns of $A$ is the vector $(0,0)$. Give an example of such a matrix $A$, or show that no such matrix exists.
- Find the projection of $\mathbf{b}$ onto the line through $\mathbf{a}$.
- $\mathbf b=(2,-1,0)$, $\mathbf a=(1,0,1)$
- $\mathbf b=(1,2,2)$, $\mathbf a=(1,1,1)$

- Let $A\mathbf{x}=\mathbf{b}$ be the inconsistent system below. Find the best approximate solution $\hat{\mathbf{x}}$ and the projection $\mathbf{p}$ of $\mathbf{b}$ to $C(A)$.
- $\begin{bmatrix}1&1\\2&0\\-1&1\end{bmatrix}\mathbf{x} =\begin{bmatrix}2\\1\\1\end{bmatrix}$
- $\begin{bmatrix}1&0\\2&1\\0&1\end{bmatrix}\mathbf{x} =\begin{bmatrix}5\\0\\0\end{bmatrix}$

- For the matrix $A$ in exercise 5(a), find the projection matrix $P$.
- When projecting $\mathbf{b}$ onto the column space of a matrix $A$, which of the four subspaces contains the projection $\mathbf{p}$? Which of the four subspaces contains the error vector $\mathbf{b}-\mathbf{p}$? Explain your answers.

- For each matrix $A$, find bases for all four fundamental subspaces.
- $A=\begin{bmatrix}1&-1&3\\-2&2&-6\end{bmatrix}$
- $A=\begin{bmatrix}1&2&4\\0&1&-2\end{bmatrix}$
- $A=\begin{bmatrix}1&3&0&-1\\2&6&1&-4\\1&3&-2&3\end{bmatrix}$
- $A=\begin{bmatrix}2&0&-1&4&2\\4&1&0&2&-1\\2&1&1&-2&-3\end{bmatrix}$

- What is the dimension of the subspace spanned by the following set? Eliminate redundant vectors from the list to reduce it to an independent set.

$\begin{bmatrix}1\\-1\\0\\0\end{bmatrix},\quad \begin{bmatrix}1\\0\\-1\\0\end{bmatrix},\quad \begin{bmatrix}1\\0\\0\\-1\end{bmatrix},\quad \begin{bmatrix}0\\1\\-1\\0\end{bmatrix},\quad \begin{bmatrix}0\\1\\0\\-1\end{bmatrix},\quad \begin{bmatrix}0\\0\\1\\-1\end{bmatrix}$ - Find bases for the column space and row space of each matrix. Which two of these four subspaces are identical?

$\begin{bmatrix}1&1&0\\1&3&1\\3&1&-1\end{bmatrix},\quad \begin{bmatrix}1&1&0\\0&2&1\\0&0&0\end{bmatrix}$ - Suppose you are given a set of four vectors in $\mathbb R^3$. Your friend tells you the set cannot possibly be independent. Is this true or false, and explain why.
- Suppose that $A$ is a $10\times10$ matrix and that its column space is contained in its null space. Explain how you know that $A$ must have $\leq5$ pivots.
- Suppose $EA=R$, with $E$ and $R$ given below. Find bases for the row space, null space, and left-null space of $A$. (Do not calculate $A$.)

$E=\begin{bmatrix}1&0&0\\6&1&0\\9&8&1\end{bmatrix},\qquad R=\begin{bmatrix}1&0&0&4\\0&1&0&3\\0&0&1&2\end{bmatrix}$ - If $A$ is a matrix that is wider than it is tall, then $A$ has a nonzero _____ space. If $A$ is a matrix that is taller than it is wide, then $A$ has a nonzero _____ space. Explain your reasoning!

- Eliminate to RREF.
- $\begin{bmatrix}-1&3&5\\-2&6&10\end{bmatrix}$
- $\begin{bmatrix}-1&3&5\\-2&6&7\end{bmatrix}$

- For each of the following matrices $A$, if it has a nonzero null space, find its null space matrix $N$ (the matrix whose columns are the special solutions). Check that $AN=0$.
- $A=\begin{bmatrix}2&3\\4&6\\-2&-3\end{bmatrix}$
- $A=\begin{bmatrix}2&3\\4&6\\-2&-6\end{bmatrix}$

- Give an example of a matrix whose column space contains the vectors $(1,1,5)$ and $(0,3,1)$ and whose null space contains the vector $(1,1,2)$.
- Going back and forth between systems and parametric forms.
- Solve the ``system’’ of equations $x - 3y - z = 0$. Write your answer in parametric vector form.
- Find the column space of the matrix $N=\begin{bmatrix}3&1\\1&0\\0&1\end{bmatrix}$

- For each system, solve or state that it is inconsistent. If the system is consistent, find the null space in parametric vector form, the particular solution, and finally write the general solution.
- $\{2y+z=1,\ x-4y-2z=1,\ x=1\}$
- $\{2y+z=1,\ x-4y-2z=2,\ x=4\}$
- $\begin{bmatrix}1&0&2&3\\1&3&2&0\\2&0&4&9\end{bmatrix}\mathbf{x} =\begin{bmatrix}2\\5\\10\end{bmatrix}$

- Give an example of a system of two equations in three variables with particular solution $\mathbf x_p=(2,4,0)$ and with special solution $(1,1,1)$.

- Find the column space of each matrix $A$. The answer consists of one or more equations in the variables $b_1,b_2,b_3$ that decide when $A\mathbf{x}=\mathbf{b}$ has a solution.
- $A=\begin{bmatrix}2&1&1\\4&2&1\\-2&-1&0\end{bmatrix}$
- $A=\begin{bmatrix}2&1&1\\4&2&2\\-2&-1&-1\end{bmatrix}$

- For each matrix below, what is the geometric shape of the column space? Your answer should say whether it is a (point / line / plane / etc) in $\mathbb R$ to the (1 / 2 / 3 / etc). Briefly explain your answers!

$\begin{bmatrix}1&2\\0&0\\0&0\end{bmatrix},\quad \begin{bmatrix}1&0\\0&2\\0&0\end{bmatrix},\quad \begin{bmatrix}1&0\\2&0\\0&0\end{bmatrix}$ - Suppose $A$ is a matrix and $B$ is the same matrix $A$ except with an additional column added onto the right. Can the column space of $B$ be larger than the column space of $A$? Can the column space of $B$ be the same as the column space of $A$? For each question give an example or say why it is not possible to do so.
- Suppose that $A$ is a $5\times5$ invertible matrix. What is the column space of $A$? How do you know?

- Factor the following matrices into $LU$ form.
- $\begin{bmatrix}2&-3\\1&-3\end{bmatrix}$
- $\begin{bmatrix}1&0&1\\2&2&2\\3&4&5\end{bmatrix}$

- Use your decomposition from the previous problem to solve each system. (Do not use elimination again! Instead rewrite each system as two triangular systems and use forth-solving and back-solving.)
- $\set{x+z=4,\ 2x+2y+2z=4,\ 3x+4y+5z=6}$
- $\set{x+z=2,\ 2x+2y+2z=6,\ 3x+4y+5z=14}$

- Factor the
*tridiagonal*matrix $A$ into $A = LU$. Do you observe anything special about the solution? $A=\begin{bmatrix}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2\end{bmatrix}$ - Consider a matrix $A$ which is already lower triangular:
$A=\begin{bmatrix}1&0&0\\a&1&0\\b&c&1\end{bmatrix}$
- What are the elimination matrices $E_1$, $E_2$, $E_3$ that reduce $A$ to upper triangular form?
- What is the product of the elimination matrices $E=E_3E_2E_1$?
- What are $L$ and $U$ in this case?

- Find the inverse of each matrix, or show that it is singular.
- $\begin{bmatrix}e&2e\\3e&4e\end{bmatrix}$
- $\begin{bmatrix}1&3\\2&7\end{bmatrix}$
- $\begin{bmatrix}1&-1&1\\1&1&1\\1&2&4\end{bmatrix}$

- If $A=\begin{bmatrix}1&1\\1&1\end{bmatrix}$ find two matrices $B,C$ such that $AB=AC$ but $B\neq C$. If $A$ is invertible, explain why this would be impossible.
- Find invertible matrices $A$ and $B$ such that $A+B$ is singular. Find singular matrices $A$ and $B$ such that $A+B$ is invertible.
- Find three values of $c$ that make the following matrix singular. Explain why it is singular in each case.

$A=\begin{bmatrix}2&c&c\\c&c&c\\8&7&c\end{bmatrix}$ - Use elimination to reduce the following matrix to upper triangular form. In your work you may assume that $a\neq 0$ and $a\neq b$. Considering the three pivots, explain why the matrix is invertible.

$A=\begin{bmatrix}a&b&b\\a&a&b\\a&a&a\end{bmatrix}$

- Let $A$ be the matrix below. Find $A^2$ and $A^3$. What will $A^{100}$ be?

$\begin{bmatrix}1&0&0\\0&1&0\\0&a&1\end{bmatrix}$ - Consider the matrices $A$ and $B$ below. Compute both $A^2+2AB+B^2$ and $(A+B)^2$. Are they the same? If not, what is the correct rule?

$A=\begin{bmatrix}1&2\\0&0\end{bmatrix},\qquad B=\begin{bmatrix}1&0\\3&0\end{bmatrix}$ - Decide whether the following statements are true or false. If true, explain why, and if false give an example showing that it is false.
- $A^2$ only makes sense if $A$ is a square matrix
- If $AB$ and $BA$ both make sense, then $A$ and $B$ are square matrices.
- If $AB$ and $BA$ both make sense, then $AB$ and $BA$ are square matrices.
- If $AB=B$ then $A$ is the identity matrix.

- For each system of equations, write the corresponding augmented matrix and use elimination to solve it. In each step, keep note of the elimination matrices you used.
- $\set{-3x+4y=-11;\ 3x+5y=20}$
- $\set{-x-2y+z=-4;\ 2x+4y+3z=3; 3y+2z=1}$

- What $3\times3$ matrix $E$ multiplies a vector $(x,y,z)$ to give $(x,y+z,2y)$? (Note: I type my vectors horizontally to save vertical space on the page only. The question will be easier if you write them vertically.)
- Apply elimination to the augmented matrix. For what value of $c$ does the system have solutions?

$\begin{bmatrix}1 & 2 & 3 &\vert &1 \\ 2 & 3 & 4 & \vert & 2 \\ 4 & 7 & 10 & \vert & c\end{bmatrix}$ - Write down the row operations corresponding to the elimination matrices:
- $\begin{bmatrix}1 & 0 & 0\\ 3 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$
- $\begin{bmatrix}1 & 2 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$
- $\begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}$

- Write down the elimination matrix that
*undoes*the affect of the given elimination matrix.- $\begin{bmatrix}1&0&0\\0&1&0\\3&0&1\end{bmatrix}$
- $\begin{bmatrix}1&0&0\\-3&1&0\\4&0&1\end{bmatrix}$ (this is a double elimination matrix which performs two operations)

- Solve the system of linear equations using any method.
- $\set{2x+y+z=2;\ x+2y+3z=1;\ y+2z=0}$

- For each of the following systems, carefully graph the equations and the location of the solution. Then graph the “vector picture” (the right-hand side as the tip of a parallelogram made from two column vectors). Finally, find the solution.
- $\set{x-2y=0;\ x+y=6}$
- $\set{2x+3y=1;\ 10x+9y=11}$

- Consider a system with two equations and three variables $x,y,z$, fill in the blanks. Explain your reasoning with one complete sentence for each answer.
- The ordinary graph of the system will show (2 / 3) (lines / planes) in (2 / 3)-dimensional space.
- The vector picture will show (2 / 3 / 4) vectors in (2 / 3)-dimensional space.

- The intersection of the two planes defined by $x+y+3z=6$ and $x-y+z=4$ is a line. Find the point on this line with $z=0$, and the point on this line with $z=2$. Find another point on this line.
- Suppose you have three planes, $P_1$, $P_2$, and $P_3$. Suppose you know that there are at least two points in the intesection of the three planes. Explain why you can conclude there are actually infinitely many points in the intersection of the three planes.

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