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Expert-verified Found in: Page 176 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Find the basis of all ${{\mathbit{P}}}_{{\mathbf{n}}}$, and determine its dimension.

The dimension of ${P}_{n}$ is $n+1$ which is spanned by role="math" localid="1659419285127" $\mathrm{Span}\left\{1,\mathrm{t},{\mathrm{t}}^{2},...,{\mathrm{t}}^{\mathrm{n}}\right\}$.

See the step by step solution

## Step 1: Determine the span

Consider the set of all polynomial ${P}_{n}$.

The set role="math" localid="1659419586773" $\left\{1,t,{t}_{2},...,{t}_{n}\right\}$is linear independent set of ${\mathbf{V}}$ if there exist constant${\mathbf{}}{{\mathbf{a}}}_{{\mathbf{i}}}{\mathbf{\in }}{\mathbf{R}}$ such that ${\mathbf{}}{{\mathbf{a}}}_{{\mathbf{0}}}{\mathbf{+}}{{\mathbf{a}}}_{{\mathbf{1}}}{{\mathbf{t}}}_{{\mathbf{1}}}{\mathbf{+}}{{\mathbf{a}}}_{{\mathbf{2}}}{{\mathbf{t}}}_{{\mathbf{2}}}{\mathbf{+}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{+}}{{\mathbf{a}}}_{{\mathbf{n}}}{{\mathbf{t}}}_{{\mathbf{n}}}{\mathbf{=}}{\mathbf{0}}$ where ${{\mathbf{a}}}_{{\mathbf{0}}}{\mathbf{=}}{{\mathbf{a}}}_{{\mathbf{1}}}{\mathbf{=}}{{\mathbf{a}}}_{{\mathbf{2}}}{\mathbf{=}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{=}}{{\mathbf{a}}}_{{\mathbf{n}}}{\mathbf{=}}{\mathbf{0}}$ .

Any polynomial $f\left(t\right)\in {P}_{n}$ is defined as follows.

$f\left(t\right)={b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+...+{b}_{n}{t}^{n}$

## Step 2: Compare the polynomials

Compare the equations${b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+...+{b}_{n}{t}^{n}=0$ bot side as follows.

$\begin{array}{l}{b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+...+.+{b}_{n}{t}^{n}=0\\ {b}_{0}+{b}_{1}t+{b}_{2}{t}^{2}+...+.+{b}_{n}{t}^{n}=\left(0\right)+\left(0\right)t+\left(0\right){t}^{2}+...+\left(0\right){t}^{n}\\ {b}_{i}=0\end{array}$

By the definition of linear independence, the subset $\left\{1,t,{t}^{2},...,{t}^{n}\right\}$is L.I where are linear independent.

Therefore, the set of polynomial ${P}_{n}$ is spanned by $Span\left\{1,t,{t}^{2},...,{t}^{n}\right\}$.

Hence, the dimension of$A$ is and $n+1$ spanned by$Span1,t,{t}^{2},...,{t}^{n}.$ ### Want to see more solutions like these? 