Question

Let P^3 ={ax^3+bx^2+cx+d ! a,b,c,d ϵ R}. Check whether f (x) = x^2+2x+1 is in[S],
the subspace of P^3 generated by S ={3x^2+1, 2x^2+x+1}.

If f (x) is in [S], write f as a linear combination of elements in S.

If f (x) is not in [S], find anotherpolynomial g(x) of degree at most two such that f (x)
is in the span of S U {g(x)}.

Alsowrite f as a linear combination of elements in S U {g(x)}.

Answer 1

Answer on Question #44929 – Math - Linear Algebra

Problem.

Let $P^3 = \{ax^3 + bx^2 + cx + d \mid a, b, c, d \in R\}$. Check whether $f(x) = x^2 + 2x + 1$ is in[S], the subspace of $P^3$ generated by $S = \{3x^2 + 1, 2x^2 + x + 1\}$.

If $f(x)$ is in [S], write $f$ as a linear combination of elements in $S$.

If $f(x)$ is not in [S], find another polynomial $g(x)$ of degree at most two such that $f(x)$ is in the span of $SU\{g(x)\}$.

Also write $f$ as a linear combination of elements in $SU\{g(x)\}$.

Solution.

$$x^2 + 2x + 1 = 2(2x^2 + x + 1) - (3x^2 + 1), \text{ so } f(x) \in [S].$$

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