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Found in: Page 191

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Consider the polynomials , and $${p_{\bf{3}}}\left( t \right) = {\bf{2}}$$ $${p_{\bf{1}}}\left( t \right) = {\bf{1}} + t,{p_{\bf{2}}}\left( t \right) = {\bf{1}} - t$$(for all t). By inspection, write a linear dependence relation among $${p_{\bf{1}}},{p_{\bf{2}}},$$ and $${p_{\bf{3}}}$$. Then find a basis for Span$$\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}$$.

The set $$\left\{ {{p_1},{p_2}} \right\}$$ is a basis for $${\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\}$$.

See the step by step solution

## Step 1: Find the relation among the polynomials

$$\begin{array}{c}{p_1}\left( t \right) + {p_2}\left( t \right) = 1 + t + 1 - t\\ = 2\\{p_1}\left( t \right) + {p_2}\left( t \right) = {p_3}\left( t \right)\end{array}$$

By the spanning set theorem, you get

$${\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\} = {\rm{Span}}\left\{ {{p_1},{p_2}} \right\}$$.

## Step 2: Use the definition of linear independence

Consider the linear combination of $${p_1}$$ and $${p_2}$$.

$$\begin{array}{c}{c_1}{p_1}\left( t \right) + {c_2}{p_2}\left( t \right) = 0\\{c_1}\left( {1 + t} \right) + {c_2}\left( {1 - t} \right) = 0\\{c_1} + {c_1}t + {c_2} - {c_2}t = 0\\\left( {{c_1} + {c_2}} \right) + \left( {{c_1} - {c_2}} \right)t = 0\end{array}$$

This implies,

$$\begin{array}{l}{c_1} + {c_2} = 0\\{c_1} - {c_2} = 0.\end{array}$$

Thus, $${c_1} = {c_2} = 0$$.

This implies $$\left\{ {{p_1},{p_2}} \right\}$$ is linearly independent.

## Step 3: Draw a conclusion

Hence, $$\left\{ {{p_1},{p_2}} \right\}$$ forms a basis for $${\rm{Span}}\left\{ {{p_1},{p_2},{p_3}} \right\}$$.