Chapter 9

Normalize \([1,3,-1]\).

Assume that the Leslie matrix is $$ L=\left[\begin{array}{ll} 0 & 2 \\ 0.6 & 0 \end{array}\right] $$ Suppose that, at time \(t=0, N_{0}(0)=5\) and \(N_{1}(0)=1\). Find the population vectors for \(t=0,1,2, \ldots, 10 .\) Compute the successive ratios $$ q_{0}(t)=\frac{N_{0}(t)}{N_{0}(t-1)} \text { and } q_{1}(t)=\frac{N_{1}(t)}{N_{1}(t-1)} $$ for \(t=1,2, \ldots, 10 .\) Do \(q_{0}(t)\) and \(q_{1}(t)\) converge? Compute the fraction of females age 0 for \(t=0,1, \ldots, 10 .\) Describe the longterm behavior of \(q_{0}(t)\).

Vectors are given in their polar coordinate representation (length \(\boldsymbol{r}\), and angle \(\alpha\) measured counterclockwise from the positive \(x_{1}-\) axis). Find the representation of the vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in Cartesian coordinates. $$r=1, \alpha=180^{\circ}$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Show that \(A+B=B+A\).

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{l} 7 x-y=4 \\ 3 x+2 y=1 \end{array} $$

Normalize \([2,0,-4]\).

Vectors are given in their polar coordinate representation (length \(\boldsymbol{r}\), and angle \(\alpha\) measured counterclockwise from the positive \(x_{1}-\) axis). Find the representation of the vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in Cartesian coordinates. $$r=5, \alpha=270^{\circ}$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Show that \((A+B)+C=A+(B+C)\).

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{l} 5 x+2 y=8 \\ -x+3 y=9 \end{array} $$

Suppose that $$ L=\left[\begin{array}{ll} 2 & 4 \\ 0.3 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Suppose a vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) has length 3 and is \(15^{\circ}\) clockwise from the positive \(x_{1}\) -axis. Find \(x_{1}\) and \(x_{2}\).

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{r} 3 x-y=1 \\ -3 x+y=4 \end{array} $$

Suppose that $$ L=\left[\begin{array}{rr} 4 & 2 \\ 1 & 0.5 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Suppose a vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) has length 2 and is \(140^{\circ}\) clockwise from the positive \(x_{1}\) -axis. Find \(x_{1}\) and \(x_{2}\).

Find the transpose of $$ A=\left[\begin{array}{l} 2 \\ 0 \\ 1 \end{array}\right] $$

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{aligned} 2 x+3 y &=5 \\ -y &=-2+\frac{2}{3} x \end{aligned} $$

Find the dot product of \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-3,-4]^{\prime}\).

Suppose that $$ L=\left[\begin{array}{ll} 3 & 2 \\ 1.5 & 1 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Suppose a vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) has length 3 and is \(25^{\circ}\) counterclockwise from the positive \(x_{2}\) -axis. Find \(x_{1}\) and \(x_{2}\).

Find the transpose of $$ A=\left[\begin{array}{rrr} -1 & 0 & 0 \\ 3 & 1 & -4 \end{array}\right] $$

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{r} x+2 y=3 \\ 4 y+2 x=6 \end{array} $$

Find the dot product of \(\mathbf{x}=[1,2]^{\prime}\) and \(\mathbf{y}=[3,-1]^{\prime}\).

Suppose that $$ L=\left[\begin{array}{ll} 0 & 5 \\ 0.9 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Suppose a vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) has length 1 and is \(90^{\circ}\) counterclockwise from the negative \(x_{2}\) -axis. Find \(x_{1}\) and \(x_{2}\).

Suppose \(A\) is a \(2 \times 2\) matrix. Find conditions on the entries of \(A\) such that $$ A-A^{\prime}=\mathbf{0} $$

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{r} x-2 y=2 \\ 4 y-2 x=-4 \end{array} $$

Find the dot product of \(\mathbf{x}=[0,-1,3]^{\prime}\) and \(\mathbf{y}=[-3,0,1]^{\prime}\).

Suppose that $$ L=\left[\begin{array}{ll} 0 & 5 \\ 0.09 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}3 \\ 0\end{array}\right]\)

Suppose that \(A\) and \(B\) are \(m \times n\) matrices. Show that $$ (A+B)^{\prime}=A^{\prime}+B^{\prime} $$

Predicting Heart Disease. The risk of a person developing heart disease is correlated with their BMI (or body mass index), which is calculated by dividing their body mass by the square of their height (high BMIs typically mean the person is overweight). Heart disease is also related to a person's level of activity. Suppose a person's BMI is \(x\) and their physical activity is \(y\) (measured, for example, in minutes of exercise each day). Let \(h\) be a heart disease risk index (high values of \(h\) mean high risk of heart disease, low values of \(h\) mean a low risk of heart disease). \(h\) depends on \(x\) and \(y\), that is, \(h=h(x, y)\). A simple linear model is that if only the effects of BMI and activity are considered: $$ h(x, y)=a x+b y $$ for some constants \(a\) and \(b\). (a) Would you expect \(a>0\) or \(a<0 ?\) What about \(b ?\) (b) To fit the values of \(a\) and \(b\), consider the following data. A patient with BMI of 20, and who does 50 min of activity each day, has risk index \(h=25\). A patient with BMI of 35, and who does 10 min of activity each day, has risk index \(h=315\). Use these data to estimate the values of \(a\) and \(b\). (c) How accurate is this model likely to be? In other words name some other factors besides physical activity and BMI that could affect \(h\), and should therefore be included in the model.

Find the dot product of \(\mathbf{x}=[2,-3,1]^{\prime}\) and \(\mathbf{y}=[3,1,-2]\).

Suppose that \(A\) is an \(m \times n\) matrix. Show that $$ \left(A^{\prime}\right)^{\prime}=A $$

Plant Growth The rate of growth of a plant depends on the amount of light available to it, which depends on whether it is growing in shade or full sun. Let the light that the plant receives be \(x, x\) could, for example, be measured in lumens (lumen is a unit for total light absorption). Growth rate also depends on how many herbivorous insects graze on the plant. Denote the number of nearby herbivorous insects by \(y(y\) could, for example, represent the number of herbivorous insects per square meter of habitat). A simple model is that the rate of growth, \(r\) is a linear function of \(x\) and \(y\); that is: $$ r(x, y)=a x+b y $$ for some constants \(a\) and \(b\). (a) Would you expect \(a>0\) or \(a<0 ?\) What sign do you expect \(b\) will have? (b) Consider the following data: A plant that receives an average of 3000 lumens of light, and that has 10 herbivorous insects per square meter, grows at a rate of \(20 \mathrm{~cm} / \mathrm{yr}\). A different plant, receiving an average of 5000 lumens of light, has 40 herbivorous insects per square meter, and grows at a rate of \(10 \mathrm{~cm} / \mathrm{yr}\). Use these data to fit \(a\) and \(b\). (c) How accurate is this model likely to be? In other words name some other factors besides light and number of herbivorous insects that could affect \(r\) and therefore should be in the model.

Use the dot product to compute the length of \([0,-1,2]\).

Suppose that \(A\) is an \(m \times n\) matrix and \(k\) is a real number. Show that $$ (k A)^{\prime}=k A^{\prime} $$

In Problems 19-24, solve each system of linear equations. $$ \begin{array}{r} 2 x-3 y+z=-1 \\ x+y-2 z=-3 \\ 3 x-2 y+z=2 \end{array} $$

Use the dot product to compute the length of \([1,1,3]\) '.

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}-1 \\ -1\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\)

Suppose that \(A\) is an \(m \times k\) matrix and \(B\) is a \(k \times n\) matrix. Show that $$ (A B)^{\prime}=B^{\prime} A^{\prime} $$

In Problems 19-24, solve each system of linear equations. $$ \begin{aligned} 5 x-y+2 z &=6 \\ x+2 y-z &=-1 \\ 3 x+2 y-2 z &=1 \end{aligned} $$

Use the dot product to compute the length of \([1,2,3,4]\) '.

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{r}-1 \\ 0\end{array}\right]\)

Let $$A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 0 \\ -1 & -1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right]$$ Compute the following: (a) \(A B\) (b) \(B A\)

In Problems 19-24, solve each system of linear equations. $$ \begin{array}{r} 2 x+y+z=7 \\ 3 x+2 y+z=9 \\ x+y-z=0 \end{array} $$

Use the dot product to compute the length of \([1,2,3,0]\).

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}-3 \\ -1\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}-1 \\ -1\end{array}\right]\)

Let $$A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 0 \\ -1 & -1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right]$$ Compute \(A B C\).

In Problems 19-24, solve each system of linear equations. $$ \begin{array}{r} -2 x+4 y-z=-1 \\ x+7 y+2 z=-4 \\ 3 x-2 y+3 z=-3 \end{array} $$

Find the angle between \(\mathbf{x}=[3,1]^{\prime}\) and \(\mathbf{y}=[3,-1]\).