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Math 511: Linear Algebra

Underdetermined and Overdetermined Systems

Underdetermined and Overdetermined Systems¶

The linear system of equations below are underdetermined because there are fewer equations than unknowns.

$$ \begin{align*} \ 3x + 4y -\ \ z &= 2 \\ -2x + 2y + 2z &= 3 \end{align*} $$

The following system of linear equations is called overdetermined because it has more equations than variables.

$$ \begin{align*} 5x_1 - 4x_2 &=\ \ 6 \\ 4x_1 + 3x_2 &= -1 \\ 3x_1 + 2x_2 &=\ \ 6 \end{align*} $$

In the following exercises if the answer is yes give an example of a linear system of the described type. Show that the solution to the linear system has the described properties. If you create a graph, include it with your pdf with your responses. If the answer is no explain why there cannot be a linear system with the described properties.

1. There are consistent underdetermined linear systems with three independent variables.¶

2. There are consistent overdetermined linear systems with three independent variables.¶

3. There are inconsistent underdetermined linear systems with three independent variables.¶

4. There are inconsistent overdetermined linear systems with three independent variables.¶

5. Explain why you would expect an overdetermined linear system to be inconsistent.¶

Must this always be the case?

6. Explain why you would expect an underdetermined linear system to have infinitely many solutions.¶

Must this always be the case?

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