Chapter 2

Linear Equations

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Problem 2.1.1

Let

A = \begin{pmatrix} 1 & 1 \\ 2 & -1 \end{pmatrix}, \qquad \overrightarrow {b} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}.

Find the solutions of the system of linear equations $A\overrightarrow {x} = \overrightarrow {b}$ .

Problem 2.1.2

Let $\alpha , b_{1}, b_{2} \in \mathbb {R}$ . Solve the system of linear equations

\begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}.

Problem 2.1.3

(i)(ii)

(i)

Let $\epsilon \in \mathbb {R}$ and

A = \begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}, \qquad \overrightarrow {b} = \begin{pmatrix} 3 \\ \epsilon \end{pmatrix}.

Find the condition on $\epsilon$ so that there is a solution of $A\overrightarrow {x} = \overrightarrow {b}$ .

(ii)

Consider the system of three linear equations

\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 10 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 1 \\ \epsilon \\ \epsilon ^{2} \end{pmatrix}

with $\epsilon \in \mathbb {R}$ . Find the condition on $\epsilon$ so that there is a solution.

Problem 2.1.4

Find all solutions of the system of linear equations

\begin{pmatrix} \cos (\theta ) & -\sin (\theta ) \\ -\sin (\theta ) & -\cos (\theta ) \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}, \qquad \theta \in \mathbb {R}

with $\overrightarrow {x} \neq \overrightarrow {0}$ . What type of equation is this?

Problem 2.1.5

Find all solutions of the system of linear equations

\begin{pmatrix} 5 & -2 & -4 \\ -2 & 2 & 2 \\ -4 & 2 & 5 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}.

Problem 2.1.6

Let $A \in \mathbb {R}^{n \times n}$ and $\overrightarrow {x}, \overrightarrow {b} \in \mathbb {R}^{n}$ . Consider the linear equation $A\overrightarrow {x} = \overrightarrow {b}$ . Show that it can be written as $\overrightarrow {x} = T\overrightarrow {x}$ , i.e. find $T\overrightarrow {x}$ .

Problem 2.1.7

If the system of linear equations $A\overrightarrow {x} = \overrightarrow {b}$ admits no solution we call the equations inconsistent. If there is a solution, the equations are called consistent. Let $A\overrightarrow {x} = \overrightarrow {b}$ be a system of $m$ linear equations in $n$ unknowns and suppose that the rank of $A$ is $m$ . Show that in this case $A\overrightarrow {x} = \overrightarrow {b}$ is consistent.

Problem 2.1.8

Find all solutions of the linear system of three equations and four unknowns

x_{1}+2 x_{2}-4 x_{3}+x_{4} = 3, \qquad 2 x_{1}-3 x_{2}+x_{3}+5 x_{4} = -4, \qquad 7 x_{1}-10 x_{3}+13 x_{4} = 0.

Problem 2.1.9

(i)(ii)

(i)

Solve the linear equation

\begin{pmatrix} x_{1} & x_{2} & x_{3} \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} x_{1} & x_{2} & x_{3} \end{pmatrix}.

(ii)

Find $x, y \in \mathbb {R}$ such that

\begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2/3 & x \\ y & 2 \end{pmatrix}.

Problem 2.1.10

Show that the curve fitting problem

\begin{array}{c|ccccc} j & 0 & 1 & 2 & 3 & 4 \\ \hline t_{j} & -1.0 & -0.5 & 0.0 & 0.5 & 1.0 \\ y_{j} & 1.0 & 0.5 & 0.0 & 0.5 & 2.0 \end{array}

by a quadratic polynomial of the form $p(t) = a_{2} t^{2}+a_{1} t+a_{0}$ leads to an overdetermined linear system.

Problem 2.1.11

Consider the overdetermined linear system $A\overrightarrow {x} = \overrightarrow {b}$ . Find an $\hat{\overrightarrow {x}}$ such that

\left\| A\hat{\overrightarrow {x}}-\overrightarrow {b}\right\| _{2} = \min _{\overrightarrow {x}} \left\| A\overrightarrow {x}-\overrightarrow {b}\right\| _{2} \equiv \min _{\overrightarrow {x}} \left\| \overrightarrow {r}(\overrightarrow {x})\right\| _{2}

with the residual vector $\overrightarrow {r}(\overrightarrow {x}) := \overrightarrow {b}-A\overrightarrow {x}$ and $\left\| \cdot \right\|_{2}$ denotes the Euclidean norm.

Problem 2.1.12

Consider the overdetermined linear system $A\overrightarrow {x} = \overrightarrow {b}$ with

A = \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \\ 1 & 5 \\ 1 & 6 \\ 1 & 7 \\ 1 & 8 \\ 1 & 9 \\ 1 & 10 \end{pmatrix}, \qquad \overrightarrow {x} = \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}, \qquad \overrightarrow {b} = \begin{pmatrix} 444 \\ 458 \\ 478 \\ 493 \\ 506 \\ 516 \\ 523 \\ 531 \\ 543 \\ 571 \end{pmatrix}.

Solve this linear system in the least squares sense (see the previous problem) by the normal equations method.

Problem 2.1.13

An underdetermined linear system is either inconsistent or has infinitely many solutions. Consider the underdetermined linear system $H\overrightarrow {x} = \overrightarrow {y}$ , where $H$ is an $n \times m$ matrix with $m>n$ and

\overrightarrow {x} = \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{pmatrix}, \qquad \overrightarrow {y} = \begin{pmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{pmatrix}.

Assume that $H\overrightarrow {x} = \overrightarrow {y}$ has infinitely many solutions. Let $Q$ be the $n \times m$ matrix

Q = \begin{pmatrix} 1 & 0 & \ldots & 0 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & & \vdots \\ 0 & 0 & \ldots & 1 & 0 & \ldots & 0 \end{pmatrix}.

We define $\hat{\overrightarrow {x}} := Q\overrightarrow {x}$ . Find

\min _{\overrightarrow {x}} \left\| Q\overrightarrow {x}-\overrightarrow {y}\right\| _{2}^{2}

subject to the constraint $\left\| H\overrightarrow {x}-\overrightarrow {y}\right\|_{2}^{2} = 0$ . Assume that $(\lambda H^{T} H+Q^{T} Q)^{-1}$ exists for all $\lambda >0$ . Apply the Lagrange multiplier method.

Problem 2.1.14

Show that solving the system of nonlinear equations with the unknowns $x_{1}, x_{2}, x_{3}, x_{4}$

\begin{aligned} (x_{1}-1)^{2}+(x_{2}-2)^{2}+x_{3}^{2} & = a^{2}(x_{4}-b_{1})^{2} \\ (x_{1}-2)^{2}+x_{2}^{2}+(x_{3}-2)^{2} & = a^{2}(x_{4}-b_{2})^{2} \\ (x_{1}-1)^{2}+(x_{2}-1)^{2}+(x_{3}-1)^{2} & = a^{2}(x_{4}-b_{3})^{2} \\ (x_{1}-2)^{2}+(x_{2}-1)^{2}+x_{3}^{2} & = a^{2}(x_{4}-b_{4})^{2} \end{aligned}

leads to a linear underdetermined system. Solve this system with respect to $x_{1}$ , $x_{2}$ and $x_{3}$ .

Problem 2.1.15

(i)(ii)

Let $A$ be an $m \times n$ matrix over $\mathbb {R}$ . We define

N_{A} := \left\{ \overrightarrow {x} \in \mathbb {R}^{n} : A\overrightarrow {x} = \overrightarrow {0} \right\} .

$N_{A}$ is called the kernel of $A$ and $\nu (A) := \dim (N_{A})$ is called the nullity of $A$ . If $N_{A}$ only contains the zero vector, then $\nu (A) = 0$ .

(i)

Let

A = \begin{pmatrix} 1 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}.

Find $N_{A}$ and $\nu (A)$ .

(ii)

Let

A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & -2 & 6 \\ -6 & 3 & -9 \end{pmatrix}.

Find $N_{A}$ and $\nu (A)$ .

Problem 2.1.16

(i)(ii)

(i)

Let $x_{1}, x_{2}, x_{3} \in \mathbb {Z}$ . Find all solutions of the system of linear equations

7 x_{1}+5 x_{2}-5 x_{3} = 8, \qquad 17 x_{1}+10 x_{2}-15 x_{3} = -42.

(ii)

Find all positive solutions.

Problem 2.1.17

Consider the inhomogeneous linear integral equation

\int _{0}^{1} (\alpha _{1}(x) \beta _{1}(y)+\alpha _{2}(x) \beta _{2}(y)) \varphi (y) \, dy+f(x) = \varphi (x) \tag {1}

for the unknown function $\varphi$ , $f(x) = x$ and

\alpha _{1}(x) = x, \qquad \alpha _{2}(x) = \sqrt{x}, \qquad \beta _{1}(y) = y, \qquad \beta _{2}(y) = \sqrt{y}.

Thus $\alpha_{1}$ and $\alpha_{2}$ are continuous in $[0,1]$ and likewise for $\beta_{1}$ and $\beta_{2}$ . We define

B_{1} := \int _{0}^{1} \beta _{1}(y) \varphi (y) \, dy, \qquad B_{2} := \int _{0}^{1} \beta _{2}(y) \varphi (y) \, dy

and

a_{\mu \nu } := \int _{0}^{1} \beta _{\mu }(y) \alpha _{\nu }(y) \, dy, \qquad b_{\mu } := \int _{0}^{1} \beta _{\mu }(y) f(y) \, dy

where $\mu , \nu = 1, 2$ . Show that the integral equation can be cast into a system of linear equations for $B_{1}$ and $B_{2}$ . Solve this system of linear equations and thus find a solution of the integral equation.

Problem 2.1.18

Consider the area-preserving map of the two-dimensional torus (modulo 1)

\begin{pmatrix} x_{1}' \\ x_{2}' \end{pmatrix} = A \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}, \qquad A = \begin{pmatrix} 4 & 15 \\ 1 & 4 \end{pmatrix}

where $\operatorname {det}\left(A\right) = 1$ (area-preserving). Consider a rational point on the torus

\begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} n_{1}/p \\ n_{2}/p \end{pmatrix}

where $p$ is a prime number (except 2, 3, 5) and $n_{1}$ , $n_{2}$ are integers between 0 and $p-1$ . One finds that the orbit has the following property. It is periodic and its period $T$ depends on $p$ alone. Consider $p = 7$ , $n_{1} = 2$ , $n_{2} = 3$ . Find the orbit and the period $T$ .

Problem 2.1.19

(i)(ii)

Let $A$ be an $m \times n$ matrix over $\mathbb {R}$ and $\overrightarrow {c}$ be an $n$ -vector in $\mathbb {R}^{n}$ .

(i)

Let

A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}.

Find out whether system (1) or system (2) has a solution.

(ii)

Let

A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}.

Find out whether system (1) or system (2) has a solution.

Remark.

Gordan's theorem tells us the following. Then exactly one of the following systems has a solution: System 1: $A\overrightarrow {x}<\overrightarrow {0}$ for some $\overrightarrow {x} \in \mathbb {R}^{n}$ . System 2: $A^{T} \overrightarrow {p} = \overrightarrow {0}$ and $\overrightarrow {p} \geq \overrightarrow {0}$ for some $\overrightarrow {p} \in \mathbb {R}^{m}$ .

Problem 2.1.20

Let

A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}, \qquad \overrightarrow {c} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.

Find out whether system (1) or system (2) has a solution.

Remark.

Farkas' theorem tells us the following. Let $A$ be an $m \times n$ matrix over $\mathbb {R}$ and $\overrightarrow {c}$ be an $n$ -vector in $\mathbb {R}^{n}$ . Then exactly one of the following systems has a solution: System 1: $A\overrightarrow {x} \leq \overrightarrow {0}$ and $\overrightarrow {c}^{T} \overrightarrow {x}>0$ for some $\overrightarrow {x} \in \mathbb {R}^{n}$ . System 2: $A^{T} \overrightarrow {y} = \overrightarrow {c}$ and $\overrightarrow {y} \geq \overrightarrow {0}$ for some $\overrightarrow {y} \in \mathbb {R}^{m}$ .

Problem 2.1.21

Let $A$ be an $n \times n$ matrix. Consider the linear equation $A\overrightarrow {x} = \overrightarrow {0}$ . If the matrix $A$ has rank $r$ , then there are $n-r$ linearly independent solutions of $A\overrightarrow {x} = \overrightarrow {0}$ . Let $n = 3$ and

A = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Find the rank of $A$ and the linearly independent solutions.

Problem 2.1.22

(i)(ii)

Consider the curve described by the equation

2 x^{2}+4 x y-y^{2}+4 x-2 y+5 = 0 \tag {1}

relative to the natural basis (standard basis $\overrightarrow {e}_{1} = \begin{pmatrix} 1 & 0 \end{pmatrix}^{T}$ , $\overrightarrow {e}_{2} = \begin{pmatrix} 0 & 1 \end{pmatrix}^{T}$ ).

(i)

Write the equation in matrix form.

(ii)

Find an orthogonal change of basis so that the equation relative to the new basis has no crossterms, i.e. no $x'y'$ term. This change of coordinate system does not change the origin.

Problem 2.1.23

Consider the $2 \times 2$ matrix

\begin{pmatrix} b & -a \\ a & b \end{pmatrix}

with $a, b \in \mathbb {R}$ and positive determinant, i.e. $a^{2}+b^{2}>0$ . Solve the equation

\begin{pmatrix} b & -a \\ a & b \end{pmatrix} \begin{pmatrix} x_{1} \\ y_{1} \end{pmatrix} = \begin{pmatrix} b & a \\ -a & b \end{pmatrix} \begin{pmatrix} x_{0} \\ y_{0} \end{pmatrix}

for the vector $\begin{pmatrix} x_{1} & y_{1} \end{pmatrix}^{T}$ with a given vector $\begin{pmatrix} x_{0} & y_{0} \end{pmatrix}^{T}$ .

Problem 2.1.24

(i)(ii)

Let $V$ be a vector space over a field $\mathbb {R}$ . Let $W$ be a subspace of $V$ . We define an equivalence relation $\sim$ on $V$ by stating that $v_{1} \sim v_{2}$ if $v_{1}-v_{2} \in W$ . The quotient space $V/W$ is the set of equivalence classes $[v]$ where $v_{1}-v_{2} \in W$ . Thus we can say that $v_{1}$ is equivalent to $v_{2}$ modulo $W$ if $v_{1} = v_{2}+w$ for some $w \in W$ . Let

V = \mathbb {R}^{2} = \left\{ \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} : x_{1}, x_{2} \in \mathbb {R} \right\}

and the subspace

W = \left\{ \begin{pmatrix} x_{1} \\ 0 \end{pmatrix} : x_{1} \in \mathbb {R} \right\} .

(i)

\begin{pmatrix} 3 \\ 0 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \begin{pmatrix} 4 \\ 1 \end{pmatrix} \sim \begin{pmatrix} -3 \\ 1 \end{pmatrix}, \qquad \begin{pmatrix} 3 \\ 0 \end{pmatrix} \sim \begin{pmatrix} 4 \\ 1 \end{pmatrix} \; ?

(ii)

Give the quotient space $V/W$ .

Problem 2.1.25

(i)(ii)

Suppose that $V$ is a vector space over a field $\mathbb {R}$ and $U \subset V$ is a subspace. We define an equivalence relation $\sim$ on $V$ by $x \sim y$ iff $x-y \in U$ . Let $V/U = V/\sim$ . Define addition and scalar multiplication on $V/U$ by $[x]+[y] = [x]+[y]$ , $c[x] = [cx]$ , where $c \in \mathbb {R}$ and

[x] = \left\{ y \in V : y \sim x \right\} .

(i)

Show that these operations do not depend on which representative $x$ we choose.

(ii)

Let $V = \mathbb {C}^{2}$ and the subspace $U = \left\{ (x_{1}, x_{2}) : x_{1} = 2 x_{2} \right\}$ . Find $V/U$ .

Problem 2.1.26

Let $b>a$ . Consider the system of linear equations

\begin{pmatrix} 1 & 1 & 1 & \ldots & 1 \\ x_{0} & x_{1} & x_{2} & \cdots & x_{n} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{0}^{n} & x_{1}^{n} & x_{2}^{n} & \ldots & x_{n}^{n} \end{pmatrix} \begin{pmatrix} w_{0} \\ w_{1} \\ w_{2} \\ \vdots \\ w_{n} \end{pmatrix} = \begin{pmatrix} b-a \\ (b^{2}-a^{2})/2 \\ (b^{3}-a^{3})/3 \\ \vdots \\ (b^{n+1}-a^{n+1})/(n+1) \end{pmatrix}.

Let $n = 2$ , $a = 0$ , $b = 1$ , $x_{0} = 0$ , $x_{1} = 1/2$ , $x_{2} = 1$ . Find $w_{0}, w_{1}, w_{2}$ .

Problem 2.1.27

Let $Y, X, A, B, C, E$ $n \times n$ matrices over $\mathbb {R}$ . Consider the system of matrix equations $Y+C E+D X = 0_{n}$ , $A E+B X = 0_{n}$ . Assume that $A$ has an inverse. Eliminate the matrix $E$ and solve the system for $Y$ .

Problem 2.1.28

(i)(ii)

For the three-body problem the following linear transformation plays a role

\begin{aligned} X(x_{1}, x_{2}, x_{3}) & = \frac{1}{3}(x_{1}+x_{2}+x_{3}) \\ x(x_{1}, x_{2}, x_{3}) & = \frac{1}{\sqrt{2}}(x_{1}-x_{2}) \\ y(x_{1}, x_{2}, x_{3}) & = \frac{1}{\sqrt{6}}(x_{1}+x_{2}-2 x_{3}). \end{aligned}

(i)

Find the inverse transformation.

(ii)

Introduce polar coordinates $x(r,\phi ) = r\sin (\phi )$ , $y(r,\phi ) = r\cos (\phi )$ and

r^{2} = \frac{1}{3} \left( (x_{1}-x_{2})^{2}+(x_{2}-x_{3})^{2}+(x_{3}-x_{1})^{2} \right).

Express $(x_{1}-x_{2})$ , $(x_{2}-x_{3})$ , $(x_{3}-x_{1})$ using these three coordinates.

Problem 2.1.29

Let $\alpha \in [0, 2\pi )$ . Find all solutions of the linear equation

\begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.

Thus $x_{1}$ and $x_{2}$ depend on $\alpha$ .

Problem 2.1.30

Consider the partial differential equation (Laplace equation)

\frac{\partial ^{2} u}{\partial x^{2}}+\frac{\partial ^{2} u}{\partial y^{2}} = 0 \quad \text{on} \quad [0,1] \times [0,1]

with the boundary conditions

u(x,0) = 1, \qquad u(x,1) = 2, \qquad u(0,y) = 1, \qquad u(1,y) = 2.

Apply the central difference scheme

\left( \frac{\partial ^{2} u}{\partial x^{2}} \right)_{j,k} \approx \frac{u_{j-1,k}-2 u_{j,k}+u_{j+1,k}}{(\Delta x)^{2}}, \qquad \left( \frac{\partial ^{2} u}{\partial y^{2}} \right)_{j,k} \approx \frac{u_{j,k-1}-2 u_{j,k}+u_{j,k+1}}{(\Delta y)^{2}}

and then solve the linear equation. Consider the cases $\Delta x = \Delta y = 1/3$ and $\Delta x = \Delta y = 1/4$ .

Problem 2.1.31

(i)(ii)

(i)

The equation of a line in the Euclidean space $\mathbb {R}^{2}$ passing through the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

(y-y_{1})(x_{2}-x_{1}) = (y_{2}-y_{1})(x-x_{1}).

Apply this equation to the points in $\mathbb {R}^{2}$ given by $(x_{1}, y_{1}) = (1, 1/2)$ , $(x_{2}, y_{2}) = (1/2, 1)$ . Consider the unit square with the corner points $(0,0)$ , $(0,1)$ , $(1,0)$ , $(1,1)$ and the map

(0,0) \to 0, \quad (0,1) \to 0, \quad (1,0) \to 0, \quad (1,1) \to 1.

We can consider this as a 2 input AND-gate. Show that the line constructed above classifies this map.

(ii)

The equation of a plane in $\mathbb {R}^{3}$ passing through the points $(x_{1}, y_{1}, z_{1})$ , $(x_{2}, y_{2}, z_{2})$ , $(x_{3}, y_{3}, z_{3})$ in $\mathbb {R}^{3}$ is given by

\operatorname {det}\left( \begin{pmatrix} x-x_{1} & y-y_{1} & z-z_{1} \\ x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{pmatrix} \right) = 0.

Apply this equations to the points $(1, 1, 1/2)$ , $(1, 1/2, 1)$ , $(1/2, 1, 1)$ . Consider the unit cube in $\mathbb {R}^{3}$ with the corner points (vertices)

(0,0,0), \quad (0,0,1), \quad (0,1,0), \quad (0,1,1), \quad (1,0,0), \quad (1,0,1), \quad (1,1,0), \quad (1,1,1)

and the map where all corner points are mapped to 0 except for $(1,1,1)$ which is mapped to 1. We can consider this as a 3 input AND-gate. Show that the plane constructed in (i) separates these solutions.

Problem 2.1.32

Find the system of linear equations for $a$ and $b$ given by

\frac{x+15}{(x+3)(x-1)} = \frac{a}{x+3}+\frac{b}{x-1}.

Solve the system of linear equations.

Problem 2.1.33

Let $k = 1, 2, 3, 4$ and $x_{0} = 1$ , $x_{5} = 0$ . Solve $x_{k-1}-2 x_{k}+x_{k+1} = 0$ .

Problem 2.1.34

Consider the polynomial $p(x) = a+bx+cx^{2}$ . Find $a, b, c$ from the conditions $p(0)=0$ , $p(1)=1$ , $p(2)=0$ .

Problem 2.1.35

(i)(ii)

Let $A \in \mathbb {C}^{n \times m}$ with $n \geq m$ and $\operatorname {rk}\left(A\right) = m$ .

(i)

Show that the $m \times m$ matrix $A^{*} A$ is invertible.

(ii)

We set $\Pi := A(A^{*} A)^{-1} A$ . Show that $\Pi$ is a projection matrix, i.e. $\Pi^{2} = \Pi$ and $\Pi = \Pi^{*}$ .

Problem 2.1.36

Let $a, b, c \in \mathbb {R}$ and $abc \neq 0$ . Find the solution of the system of linear equations

\begin{pmatrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \end{pmatrix} \begin{pmatrix} \cos (\alpha ) \\ \cos (\beta ) \\ \cos (\gamma ) \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}.

Problem 2.1.37

Let $A$ be an invertible $n \times n$ matrix over $\mathbb {R}$ . Consider the system of linear equation $A\overrightarrow {x} = \overrightarrow {b}$ or

\sum _{j=1}^{n} a_{ij} x_{j} = b_{i}, \qquad i = 1, \ldots , n.

Let $A = C-R$ . This is called a splitting of the matrix $A$ and $R$ is the defect matrix of the splitting. Consider the iteration

C \overrightarrow {x}^{(k+1)} = R \overrightarrow {x}^{(k)}+\overrightarrow {b}, \qquad k = 0, 1, 2, \ldots .

Let

A = \begin{pmatrix} 4 & -1 & 0 \\ -1 & 4 & -1 \\ 0 & -2 & 4 \end{pmatrix}, \qquad C = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix}, \qquad \overrightarrow {b} = \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}, \qquad \overrightarrow {x}^{(0)} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.

The iteration converges if $\rho (C^{-1} R)<1$ , where $\rho (C^{-1} R)$ denotes the spectral radius of $C^{-1} R$ . Show that $\rho (C^{-1} R)<1$ . Perform the iteration.

Problem 2.1.38

(i)(ii)

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$ and let $\overrightarrow {b} \in \mathbb {R}^{n}$ . Consider the linear equation $A\overrightarrow {x} = \overrightarrow {b}$ . Assume that $a_{jj} \neq 0$ for $j = 1, \ldots , n$ . We define the diagonal matrix $D = \operatorname {diag}(a_{jj})$ . Then the linear equation $A\overrightarrow {x} = \overrightarrow {b}$ can be written as

\overrightarrow {x} = B\overrightarrow {x}+\overrightarrow {c}

with $B := -D^{-1}(A-D)$ , $\overrightarrow {c} := D^{-1} \overrightarrow {b}$ . The Jacobi method for the solution of the linear equation $A\overrightarrow {x} = \overrightarrow {b}$ is given by

\overrightarrow {x}^{(k+1)} = B \overrightarrow {x}^{(k)}+\overrightarrow {c}, \qquad k = 0, 1, \ldots

where $\overrightarrow {x}^{(0)}$ is any initial vector in $\mathbb {R}^{n}$ . The sequence converges if

\rho (B) := \max _{j=1,\ldots ,n} \left|\lambda _{j}(B)\right| < 1

where $\rho (B)$ is the spectral radius of $B$ . Let

A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix}.

(i)

Show that the Jacobi method can applied for this matrix.

(ii)

Find the solution of the linear equation with $\overrightarrow {b} = \begin{pmatrix} 1 & 1 & 1 \end{pmatrix}^{T}$ .

Problem 2.1.39

Kirchhoff’s current law states that the algebraic sum of all the currents flowing into a junction is 0. Kirchhoff’s voltage law states that the algebraic sum of all the voltages around a closed circuit is 0. Use Kirchhoff's laws and Ohm’s law ( $V = RI$ ) to setting up the system of linear equations for the circuit depicted in the figure.

Circuit for Problem 2.1.39

Given the voltages $V_{A}$ , $V_{B}$ and the resistors $R_{1}$ , $R_{2}$ , $R_{3}$ . Find $I_{1}$ , $I_{2}$ , $I_{3}$ .

Problem 2.1.40

Write a C++ program that implements Gauss elimination to solve linear equations. Apply it to the system

\begin{pmatrix} 1 & 1 & 1 \\ 8 & 4 & 2 \\ 27 & 9 & 3 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 1 \\ 5 \\ 14 \end{pmatrix}.

Section 2.2Supplementary Problems[7/0%]

Problem 2.2.1

Let $\overrightarrow {n}$ and $\overrightarrow {p}$ be vectors in $\mathbb {R}^{n}$ with $\overrightarrow {n} \neq \overrightarrow {0}$ . The set of all vectors $\overrightarrow {x}$ in $\mathbb {R}^{n}$ which satisfy the equation $\overrightarrow {n} \cdot (\overrightarrow {x}-\overrightarrow {p}) = 0$ is called a hyperplane through the point $\overrightarrow {p} \in \mathbb {R}$ . We call $\overrightarrow {n}$ a normal vector for the hyperplane and call $\overrightarrow {n} \cdot (\overrightarrow {x}-\overrightarrow {p}) = 0$ a normal equation for the hyperplane. Find $\overrightarrow {n}$ and $\overrightarrow {p}$ in $\mathbb {R}^{4}$ such that we obtain the hyperplane given by

x_{1}+x_{2}+x_{3}+x_{4} = \frac{7}{2}.

Any hyperplane of the Euclidean space $\mathbb {R}^{n}$ has exactly two unit normal vectors.

Problem 2.2.2

The hyperplanes $x_{1}+x_{2}+x_{3}+x_{4} = 1$ , $x_{1}+x_{2}+x_{3}+x_{4} = 2$ in $\mathbb {R}^{4}$ do not intersect. Find the shortest distance.

Problem 2.2.3

(i)(ii)

(i)

Find the solutions of the equation

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.

(ii)

Find the solutions of the equation

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

Problem 2.2.4

Let $A$ be a given $3 \times 3$ matrix over $\mathbb {R}$ with $\operatorname {det}\left(A\right) \neq 0$ . Is the transformation

x_{1}'(x_{1}, x_{2}) = \frac{a_{11} x_{1}+a_{12} x_{2}+a_{13}}{a_{31} x_{1}+a_{32} x_{2}+a_{33}}, \qquad x_{2}'(x_{1}, x_{2}) = \frac{a_{21} x_{1}+a_{22} x_{2}+a_{23}}{a_{31} x_{1}+a_{32} x_{2}+a_{33}}

invertible? If so find the inverse.

Problem 2.2.5

(i)(ii)(iii)

(i)

Consider the linear equation written in matrix form

\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}.

The determinant of the $3 \times 3$ matrix is nonzero ( $+1$ ) and the inverse matrix is

\begin{pmatrix} 2 & 0 & -1 \\ 1 & 1 & -1 \\ -1 & 0 & 1 \end{pmatrix}.

Apply two different methods (Gauss elimination and the Leverrier's method) to find the solution. Compare the two methods and discuss.

(ii)

Apply the Gauss-Seidel method to solve the linear system

\begin{pmatrix} 4 & -1 & -1 & 0 \\ -1 & 4 & 0 & -1 \\ -1 & 0 & 4 & -1 \\ 0 & -1 & -1 & 4 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}

and show that the solution is given by

x_{1} = \frac{7}{24}, \qquad x_{2} = \frac{1}{12}, \qquad x_{3} = \frac{1}{12}, \qquad x_{4} = \frac{1}{24}.

(iii)

Show that the solution of the system of linear equations

x_{1}+x_{2}+x_{3} = 0, \qquad x_{1}+2 x_{2}+x_{3} = 1, \qquad 2 x_{1}+x_{2}+x_{3} = 2

is given by $x_{1} = 2$ , $x_{2} = 1$ , $x_{3} = -3$ .

Problem 2.2.6

(i)(ii)

(i)

Given the $2 \times 2$ matrix $A$ . Find all $2 \times 2$ matrices $X$ such that $AX = XA$ . From $AX = XA$ we obtain

\begin{pmatrix} a_{11} x_{11}+a_{12} x_{21} & a_{11} x_{12}+a_{12} x_{22} \\ a_{21} x_{11}+a_{22} x_{21} & a_{21} x_{12}+a_{22} x_{22} \end{pmatrix} = \begin{pmatrix} a_{11} x_{11}+a_{21} x_{12} & a_{12} x_{11}+a_{22} x_{12} \\ a_{11} x_{21}+a_{21} x_{22} & a_{12} x_{21}+a_{22} x_{22} \end{pmatrix}.

Thus we obtain the four equations

a_{12} x_{21} = a_{21} x_{12}, \qquad a_{11} x_{12}+a_{12} x_{12} = a_{12} x_{11}+a_{22} x_{12}

a_{21} x_{12} = a_{12} x_{21}, \qquad a_{21} x_{11}+a_{22} x_{21} = a_{11} x_{21}+a_{21} x_{22}.

(ii)

Show that the solution of the equation

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

is given by $x_{11} = x_{22}$ , $x_{12} = x_{21}$ .

Problem 2.2.7

Let $\alpha , b_{1}, b_{2} \in \mathbb {R}$ . Solve the system of linear equations

\begin{pmatrix} \cosh (\alpha ) & \sinh (\alpha ) \\ \sinh (\alpha ) & \cosh (\alpha ) \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}.