Question

Complete the set S = {x
3 +x
2 +1, x
2 +x+1, x+1} to get a basis of P3

Answer 1

Answer on Question #81142 – Math – Linear Algebra

Question

Complete the set $S = \{x$

3 + x

2 + 1, x

2 + x + 1, x + 1 \} to get a basis of P3

Solution

Denote

$$a_1 = x^3 + x^2 + 1, a_2 = x^2 + x + 1, a_3 = x + 1.$$

Let's check the hypothesis that $a_1, a_2, a_3$ and $a_4 = 1$ form a basis in $P_3$. We have to check whether $c_1a_1 + c_2a_2 + c_3a_3 + c_4a_4 = 0$

yields

$$c_1 = c_2 = c_3 = c_4 = 0.$$

We have

$$c_1(x^3 + x^2 + 1) + c_2(x^2 + x + 1) + c_3(x + 1) + c_4 \cdot 1 = 0$$

This means

$$c_1x^3 + (c_1 + c_2)x^2 + (c_2 + c_3)x + (c_1 + c_2 + c_3 + c_4) = 0.$$

Then

$$\left\{ \begin{array}{l} c_1 = 0 \\ c_1 + c_2 = 0 \\ c_2 + c_3 = 0 \\ c_1 + c_2 + c_3 + c_4 = 0 \end{array} \right.$$

from which

$$c_1 = 0, c_2 = -c_1 = 0, c_3 = -c_2 = 0, c_4 = -c_1 - c_2 - c_3 = 0.$$

This means that $a_1, a_2, a_3, a_4$ really form a basis in $P_3$.

Answer: 1 should be added, the basis of $P_3$ is $\{x^3 + x^2 + 1, x^2 + x + 1, x + 1, 1\}$.

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