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## Homework Statement

Do the polynomials [itex]t^{3} + 2t + 1,t^{2} - t + 2, t^{3} +2, -t^{3} + t^{2} - 5t + 2[/itex] span [itex]P_{3}[/itex]?

## Homework Equations

N/A

## The Attempt at a Solution

My attempt: let [itex]at^{3} + bt^{2} + ct + d[/itex] be an arbitrary vector in [itex]P_{3}[/itex], then:

[itex]c_{1}(t^{3} + 2t + 1) + c_{2}(t^{2} - t + 2) + c_{3}(t^{3} +2) + c_{4}(-t^{3} + t^{2} - 5t + 2) = at^{3} + bt^{2} + ct + d[/itex]

[itex]c_{1} + c_{3} - c_{4} = a[/itex]

[itex] c_{2} + c_{4} = b[/itex]

[itex]2c_{1} - c_{2} - 5c_{4} = c[/itex]

[itex]c_{1} + 2c_{2} + 2c_{3} + 2c_{4} = d[/itex]

Which becomes the augmented matrix:

[itex]\left[\begin{array}{cccc|c}1&0&1&-1&a\\0&1&0&1&b\\2&-1&0&-5&c\\1&2&2&2&d\end{array}\right][/itex]

Row reducing it, I come to the matrix:

[itex]\left[\begin{array}{cccc|c}1&0&1&-1&a\\0&1&0&1&b\\0&0&1&1&d-a-2b\\0&0&0&0&c-4a-3b+2d\end{array}\right][/itex]

Although the book did a different last row operation in the solution, I came to the same conclusion as they did except they had -2 in the 3x3 and 3x4 position rather than 1 as in my solution. Anyways, the book concludes the system is inconsistent regardless of specific values chosen for a, b, c, and d. But, wouldn't the system be consistent if a = b = c = d = 0? Then the fourth row would be [ 0 0 0 0 | 0], correct?