- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
1.10 Exercises
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In the linear space of all real polynomials $p(t),$ describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace:
(a) $\{1, t^2, t^4\};$ (b) $\{t, t^3, t^5\};$ (c) $\{t, t^3\};$ (d) $\{1 + t, (1 + t)^2\}.$
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The dimension of a subspace is given by the number of elements in its basis.
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$\text{(a)}\quad$ The subspace of even polynomials of degree $\leq 4.$ Dimension $3.$
$\text{(b)}\quad$ The subspace of odd polynomials of degree $\leq 5.$ Dimension $3.$
$\text{(c)}\quad$ The subspace of odd polynomials of degree $\leq 3.$ Dimension $2.$
$\text{(d)}\quad$ Each polynomial in the subspace of real polynomials $p(t)$ spanned by $\{1 + t, (1 + t)^2\}$ has the form
\begin{align*}
f(t) &= a_0(1 + t) + a_1(1+t)^2
\end{align*}
where $a_0$ and $a_1$ are real coefficients. From this, we can see that the subspace is the set of polynomials such that $f(-1) = 0.$ Dimension $2.$