Math Problems Vault

Problem 2.4.1

(a)(b)(c)(d)

Determine the general solution for each of the following homogeneous systems.

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(a)

$\begin{aligned} [t] x_{1} + 2 x_{2} + x_{3} + 2 x_{4} & = 0, \\ 2 x_{1} + 4 x_{2} + x_{3} + 3 x_{4} & = 0, \\ 3 x_{1} + 6 x_{2} + x_{3} + 4 x_{4} & = 0. \end{aligned}$

(b)

$\begin{aligned} [t] 2x + y + z & = 0, \\ 4x + 2y + z & = 0, \\ 6x + 3y + z & = 0, \\ 8x + 4y + z & = 0. \end{aligned}$

(c)

$\begin{aligned} [t] x_{1} + x_{2} + 2 x_{3} & = 0, \\ 3 x_{1} + 3 x_{3} + 3 x_{4} & = 0, \\ 2 x_{1} + x_{2} + 3 x_{3} + x_{4} & = 0, \\ x_{1} + 2 x_{2} + 3 x_{3} - x_{4} & = 0. \end{aligned}$

(d)

$\begin{aligned} [t] 2x + y + z & = 0, \\ 4x + 2y + z & = 0, \\ 6x + 3y + z & = 0, \\ 8x + 5y + z & = 0. \end{aligned}$

Problem 2.4.2

Among all solutions that satisfy the homogeneous system

\begin{aligned} x + 2y + z & = 0, \\ 2x + 4y + z & = 0, \\ x + 2y - z & = 0, \end{aligned}

determine those that also satisfy the nonlinear constraint $y - xy = 2z$ .

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Problem 2.4.3

Consider a homogeneous system whose coefficient matrix is

A = \begin{bmatrix} 1 & 2 & 1 & 3 & 1 \\ 2 & 4 & -1 & 3 & 8 \\ 1 & 2 & 3 & 5 & 7 \\ 2 & 4 & 2 & 6 & 2 \\ 3 & 6 & 1 & 7 & -3 \end{bmatrix}.

First transform $A$ to an unreduced row echelon form to determine the general solution of the associated homogeneous system. Then reduce $A$ to $E_{A}$ , and show that the same general solution is produced.

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Problem 2.4.4

If $A$ is the coefficient matrix for a homogeneous system consisting of four equations in eight unknowns and if there are five free variables, what is $\operatorname {rk}\left(A\right)$ ?

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Problem 2.4.5

(a)(b)

Suppose that $A$ is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that $A$ has at least one nonzero row.

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(a)

Determine the fewest number of free variables that are possible.

(b)

Determine the maximum number of free variables that are possible.

Problem 2.4.6

Explain why a homogeneous system of $m$ equations in $n$ unknowns where $m < n$ must always possess an infinite number of solutions.

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Problem 2.4.7

Construct a homogeneous system of three equations in four unknowns that has

x_{2} \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_{4} \begin{bmatrix} -3 \\ 0 \\ 2 \\ 1 \end{bmatrix}

as its general solution.

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Problem 2.4.8

If $\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ are columns that represent two particular solutions of the same homogeneous system, explain why the sum $\mathbf{c}_{1} + \mathbf{c}_{2}$ must also represent a solution of this system.

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