§ 6

A system of linear equations with matrix $A$ is consistent for any column of constant terms if and only if the rows of the matrix $A$ are linearly independent. Prove this

The vectors $\mathbf{a}$ and $\mathbf{b}$, $\mathbf{a} \neq \mathbf{0}$, are given. By means of the Fredholm theorem prove that the equation $[\mathbf{a}, \mathbf{x}]=\mathbf{b}$ has a solution if and only if $(\mathbf{a}, \mathbf{b})=0$.

Find a fundamental matrix for the system with matrix $\left\| \begin{array}{lll}1 & 1 & 1\end{array}\right\|$.

Let $\left\| E_{r} \mid B\right\|$ be the reduced form of the matrix of a homogeneous system of equations with $n$ unknowns. Find a fundamental matrix of the system.

Let $F$ be a fundamental matrix of a system of linear equations $A \mathbf{x}=\mathbf{o}$ with $n$ unknowns and let the rows of $A$ be linearly independent. What will be a fundamental matrix of the system

Write the general solution of the system with augmented matrix

Let a matrix $F$ of size $n \times p$ be a fundamental matrix of some system of equations. Prove that $F^{\prime }$ will be a fundamental matrix of the same system if and only if there exists a nonsingular matrix $Q$ of order $p$, such that $F^{\prime }=F Q$.

A system of three equations in two unknowns is considered. Verify that the application of the Fredholm theorem to this system is equivalent to the following (geometrically obvious) statement: the vector $\mathbf{b}$ expands in the vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ if and only if it is orthogonal to every vector $\mathbf{y}$ orthogonal to these vectors.

Let the system of equations $A \mathbf{x}=\mathbf{b}$ be consistent. Prove that the equation $\mathbf{a x}=\beta$ is a consequence of this system if and only if it is a linear combination of its equations.