Chapter 9

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}-2 \\ 1\end{array}\right]\) and \(a=2\)

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

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

Find the angle between \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-2,4]\).

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}3 \\ -1\end{array}\right]\) and \(a=-1\)

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

In Problems 19-24, solve each system of linear equations. $$ \begin{array}{r} 2 x+y-2 z=3 \\ 2 x-3 y-4 z=0 \\ x-5 y+3 z=-6 \end{array} $$

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

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}0 \\ -4\end{array}\right]\) and \(a=0.5\)

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

In Problems 25-28, find the augmented matrix and use it to solve the system of linear equations. $$ \begin{array}{lr} 3 x-2 y+z= & 4 \\ 4 x+y-2 z= & -12 \\ 2 x-3 y+z= & 7 \end{array} $$

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

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}3 \\ -9\end{array}\right]\) and \(a=-1 / 3\)

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

In Problems 25-28, find the augmented matrix and use it to solve the system of linear equations. $$ \begin{aligned} -x-2 y+3 z &=-9 \\ 2 x+y-z &=5 \\ 4 x-3 y+5 z &=-9 \end{aligned} $$

Let \(\mathbf{x}=[1,1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{r}-4 \\ 1\end{array}\right]\) and \(a=1 / 4\)

Suppose that \(A\) is a \(3 \times 4\) matrix and \(B\) is a \(4 \times 2\) matrix. What is the size of the product \(A B\) ?

In Problems 25-28, find the augmented matrix and use it to solve the system of linear equations. $$ \begin{array}{l} y+x=3 \\ z-y=-1 \\ x+z=2 \end{array} $$

Let \(\mathbf{x}=[-2,1]\) '. Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

Compute ax for each vector \(\mathbf{x}\) and scalar \(a\). Represent \(\mathbf{x}\) and ax in the plane, and explain graphically how you obtain \(a \mathrm{x}\) \(\mathbf{x}=\left[\begin{array}{l}0.5 \\ 0.25\end{array}\right]\) and \(a=4\)

Suppose \(A\) is a \(3 \times 4\) matrix and \(B\) is an \(m \times n\) matrix. What are values of \(m\) and \(n\) such that the following products are defined? (a) \(A B\) (b) \(B A\)

In Problems 25-28, find the augmented matrix and use it to solve the system of linear equations. $$ \begin{aligned} 2 x+z &=4 y-1 \\ x+2 y+9 &=3 z \\ 3 x+2 z &=4-2 y \end{aligned} $$

Let \(\mathbf{x}=[3,-2,1]^{\prime}\). Find any vector \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular. [Your solution will not be unique.]

Suppose that \(A\) is a \(3 \times 4\) matrix, \(B\) is a \(1 \times 3\) matrix, \(C\) is a \(3 \times 1\) matrix, and \(D\) is a \(4 \times 3\) matrix. Which of the matrix multiplications that follow are defined? Whenever it is defined, state the size of the resulting matrix. (a) \(B D^{\prime}\) (b) \(D A\) (c) \(A C B\)

Let \(\mathbf{x}=[2,0,1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{l} x-y=2 \\ x+y+z=3 \end{array} $$

A triangle has vertices at coordinates \(P=(0,0), Q=(0,3)\), and \(R=(4,3)\). (a) Use basic trigonometry to compute the lengths of all three sides and the measures of all three angles. (b) Use the results of this section to repeat (a).

Let \(A=\left[\begin{array}{rr}1 & 3 \\ 0 & -2\end{array}\right]\) and \(B=\left[\begin{array}{rrrr}1 & 0 & 0 & -3 \\ 2 & 1 & -1 & 0\end{array}\right]\) (a) Compute \(A B\). (b) Compute \(B^{\prime} A\).

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 2 x-y=3 \\ x-y=4 \\ x-3 y=1 \end{array} $$

A triangle has vertices at coordinates \(P=(0,0), Q=(0,3)\), and \(R=(2,0)\). (a) Use basic trigonometry to compute the lengths of all three sides and the measures of all three angles. (b) Use the results of this section to repeat (a).

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(\mathbf{v}-\frac{1}{2} \mathbf{u}\) and illustrate the result graphically.

Let \(A=\left[\begin{array}{lll}1 & 4 & -2\end{array}\right]\) and \(B=\left[\begin{array}{r}-1 \\ 2 \\ 2\end{array}\right]\) (a) Compute \(A B\). (b) Compute \(B A\).

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 4 y-3 z=6 \\ 2 y+z=1 \\ y+z=0 \end{array} $$

A triangle has vertices at coordinates \(P=(1,2,3), Q=\) \((1,1,2)\), and \(R=(4,2,2)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(\mathbf{u}+\mathbf{v}+\mathbf{w}\) and illustrate the result graphically.

Let $$A=\left[\begin{array}{rr} 2 & 1 \\ 0 & -3 \end{array}\right]$$ Find \(A^{2}, A^{3}\), and \(A^{4}\).

A triangle has vertices at coordinates \(P=(2,1,5), Q=\) \((-1,3,7)\), and \(R=(2,-4,1)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(2 \mathbf{v}-\mathbf{w}\) and illustrate the result graphically.

Suppose that $$A=\left[\begin{array}{rr} 1 & -1 \\ 3 & 0 \\ 5 & 2 \end{array}\right] \text { and } B=\left[\begin{array}{lll} 2 & 3 & 1 \\ 1 & 0 & 0 \end{array}\right]$$ Show that \((A B)^{\prime}=B^{\prime} A^{\prime}\).

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{rr} x+y= & -1 \\ 2 x-y= & 7 \\ x-2 y= & 8 \end{array} $$

Find the equation of the line through \((2,1)\) and perpendicular to \([1,2]^{\prime}\).

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$$

Let $$B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$ (a) Find \(B^{2}, B^{3}, B^{4}\), and \(B^{5}\). (b) What can you say about \(B^{k}\) when (i) \(k\) is even and (ii) \(k\) is odd?

SplendidLawn sells three types of lawn fertilizer: SL 24-4- 8, SL 21-7-12 and SL \(17-0-0 .\) The three numbers refer to the percentages of nitrogen, phosphate, and potassium, in that order, of the contents. (For instance, \(100 \mathrm{~g}\) of SL 24-4-8 contains \(24 \mathrm{~g}\) of nitrogen.) Suppose that each year your lawn requires \(500 \mathrm{~g}\) of nitrogen, \(100 \mathrm{~g}\) of phosphate, and \(180 \mathrm{~g}\) of potassium. How much of each of the three types of fertilizer should you apply?

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\left[\begin{array}{rr}2 & 0 \\ 0 & -1\end{array}\right]$$

Let $$I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$ Show that \(I_{3}=I_{3}^{2}=I_{3}^{3}\).

Microbiological Diversity DNA sequencing allows the different bacteria and fungi present in a patch of soil to be identified. Many new species have been found by this method, and it also reveals the diversity of microorganism ecosystems. You are an ecologist explaining differences in diversity between different soil habitats. (We have previously met the GiniSimpson diversity index and Shannon diversity index as ways of quantifying diversity.) You believe that diversity is affected by the amount of rainfall (because rain introduces new microbes into the soil, and also leads to ground-water flows that redistribute microbes). Also, some microbes (e.g., Streptomyces bacteria) produce antibiotics that can suppress the growth of other microbes, reducing the overall diversity. Let \(x\) be the amount of rainfall (measured, e.g., in \(\mathrm{mm} /\) day \()\) and \(y\) be the number of antibiotic-producing species that are present. Then you hypothesize that diversity \(d\) is a linear function of \(x\) and \(y\) : $$ d(x, y)=a x+b y+c $$ where \(a, b, c\) are all constants. (a) Explain why including the constant \(c\) allows \(d\) to be non-zero even when \(x=0\) and \(y=0 .\) Does that make sense biologically? (b) Do you expect \(a>0\) or \(a<0 ?\) What sign do you expect \(b\) to have? (c) Use the following data to fit the parameters \(a, b\), and \(c\). A patch of soil with \(3 \mathrm{~mm} /\) day average rainfall, and no antibiotic-producing species has diversity \(d=0.65\). A patch of soil with \(5 \mathrm{~mm} /\) day average rainfall, and 10 antibiotic-producing species has diversity \(d=0.65\). A patch of soil with \(1 \mathrm{~mm} /\) day average rainfall, and 5 antibiotic-producing species has diversity \(d=0.5\).

Find the equation of the line through \((1,-2)\) and perpendicular to \([4,1]^{\prime}\).

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}\right]$$