Math Problems Vault

Problem 7.3.1

Determine all solutions to the following infinite system of linear equations in the infinitely many unknowns $x_{1}, x_{2}, \ldots$ :

\begin{align} x_{1}+x_{3}+x_{5} & = 0 \\ x_{2}+x_{4}+x_{6} & = 0 \\ x_{3}+x_{5}+x_{7} & = 0 \\ & \ \vdots \end{align}

How many free parameters are required?

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Problem 7.3.2

(1)(2)

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(1)

Using only the axioms for a field $\mathbf{F}$ , prove that a system of $m$ homogeneous linear equations in $n$ unknowns with $m < n$ and coefficients in $\mathbf{F}$ has a nonzero solution.

(2)

Use Part 1 to show that if $V$ is a vector space over $\mathbf{F}$ which is spanned by a finite number of elements, then every maximal linearly independent subset of $V$ has the same number of elements.

Problem 7.3.3

If a finite homogeneous system of linear equations with rational coefficients has a nontrivial complex solution, need it have a nontrivial rational solution? Give a proof or a counterexample.

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Problem 7.3.4

Let $A$ be a real $m \times n$ matrix with rational entries and let $b$ be an $m$ -tuple of rational numbers. Assume that the system of equations $Ax = b$ has a solution $x$ in complex $n$ -space $\mathbb {C}^{n}$ . Show that the equation has a solution vector with rational components, or give a counterexample.

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