Math Problems Vault

Problem 19.13

Prove that if the columns of the main matrix are linearly independent, then the system of linear equations has at most one solution.

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Problem 19.14

Prove that if the rows of the main matrix are linearly independent, then the system of equations is consistent for any column of constant terms.

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Problem 19.15

Prove the following statement: if a system of linear equations is consistent for any column of constant terms, then the rows of its main matrix are linearly independent.

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Problem 19.16

Prove that exactly one of two possibilities holds: either the system of linear equations $A\mathbf{x} = \mathbf{b}$ is consistent for any column of constant terms, or its adjoint homogeneous system $A^{T} \mathbf{y} = 0$ has a nonzero solution (the Fredholm alternative).

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Problem 19.17

(1)(2)(3)(4)

Formulate conditions (and prove their necessity and sufficiency) that the main matrix must satisfy so that the number of solutions of the system of linear equations, depending on the column $b$ of constant terms, equals:

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(1)

0 or 1

(2)

1 or $\infty$

(3)

0 or $\infty$

(4)

1 for all b.

Problem 19.18

(1)(2)(3)

A system of linear equations is given by its augmented matrix. Check the consistency of this system, using the Fredholm theorem and the result of problem 18.9 for the corresponding adjoint system of equations:

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(1)

$\left\| \mathbf{A}_{114} \mid \mathbf{c}_{89}\right\|$ . $\mathbf{A}_{114} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \\ 4 & 7 \end{smallmatrix}\right], \mathbf{c}_{89} = \left[\begin{smallmatrix} 1 \\ 2 \\ -1 \end{smallmatrix}\right]$ .

(2)

$\left\| \mathbf{A}_{118} \mid \mathbf{c}_{90}\right\|$ . $\mathbf{A}_{118} = \left[\begin{smallmatrix} 3 & 4 \\ 5 & 7 \\ 2 & 3 \end{smallmatrix}\right], \mathbf{c}_{90} = \left[\begin{smallmatrix} 2 \\ 3 \\ 1 \end{smallmatrix}\right]$ .

(3)

$\left\| \mathbf{A}_{587} \mid \mathbf{c}_{171}\right\|$ . $\mathbf{A}_{587} = \left[\begin{smallmatrix} 1 & 2 & 2 & 5 & 3 \\ 2 & 3 & -5 & 26 & -4 \\ 3 & -4 & 8 & -9 & 1 \\ -4 & 1 & -3 & -12 & 2 \end{smallmatrix}\right], \mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right]$ .

Problem 19.19

(1)(2)(3)

A system of equations is given by its augmented matrix containing a parameter. Using the Fredholm theorem, find all values of the parameter for which the system is consistent, and solve it:

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(1)

$\left\| \mathbf{A}_{205} \mid \mathbf{c}_{91}\right\|$ . $\mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right]$ .

(2)

$\left\| \mathbf{A}_{515} \mid \mathbf{c}_{91}\right\|$ . $\mathbf{A}_{515} = \left[\begin{smallmatrix} 3 & -2 & 5 & 4 \\ 6 & -4 & 4 & 3 \\ 9 & -6 & 3 & 2 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right]$ .

(3)

$\left\| \mathbf{A}_{507} \mid \mathbf{c}_{91}\right\|$ . $\mathbf{A}_{507} = \left[\begin{smallmatrix} 1 & 3 & 4 & 1 \\ 2 & 2 & 3 & 1 \\ 4 & 0 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{91} = \left[\begin{smallmatrix} \alpha \\ \alpha^{2} \\ \alpha^{3} \end{smallmatrix}\right]$ .

Problem 19.20

A system of equations is given by its augmented matrix $\left\| \mathbf{A}_{644} \mid \mathbf{c}_{282}\right\|$ , depending on the parameters $\lambda_{1}, \ldots , \lambda_{n}, \mu$ . Describe the set of values of the parameters for which the system is consistent, and solve it. $\mathbf{A}_{644} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ \lambda_1 & \lambda_2 & \lambda_3 & \ldots & \lambda_n \\ \lambda_1^2 & \lambda_2^2 & \lambda_3^2 & \ldots & \lambda_n^2 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \ldots \\ \lambda_1^{n-1} & \lambda_2^{n-1} & \lambda_3^{n-1} & \ldots & \lambda_n^{n-1} \end{smallmatrix}\right], \mathbf{c}_{282} = \left[\begin{smallmatrix} 1 \\ \mu \\ \mu^2 \\ \vdots \\ \mu^{n-1} \end{smallmatrix}\right]$ .

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