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Expert-verified 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 # In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.$${\bf{1}} - {\bf{2}}{t^{\bf{2}}} - {t^{\bf{3}}}$$, $$t + {\bf{2}}{t^{\bf{3}}}$$, $${\bf{1}} + t - {\bf{2}}{t^{\bf{2}}}$$

The polynomials are linearly independent.

See the step by step solution

## Step 1: Write the polynomials in the standard vector form

The vectors of the given polynomials can be written as follows:

$$1 - 2{t^2} - {t^3} \equiv \left( {\begin{array}{*{20}{c}}1\\0\\{ - 2}\\{ - 1}\end{array}} \right)$$, $$t + 2{t^3} \equiv \left( {\begin{array}{*{20}{c}}0\\1\\0\\2\end{array}} \right)$$, $$1 + t - 2{t^2} \equiv \left( {\begin{array}{*{20}{c}}1\\1\\{ - 2}\\0\end{array}} \right)$$

## Step 2: Form the matrix using the vectors

The matrix formed by using the vectors of the polynomials is:

$$A = \left( {\begin{array}{*{20}{c}}1&0&1\\0&1&1\\{ - 2}&0&{ - 2}\\{ - 1}&2&0\end{array}} \right)$$

## Step 3: Write the matrix in the echelon form

$$\left( {\begin{array}{*{20}{c}}1&0&1\\0&1&1\\{ - 2}&0&{ - 2}\\{ - 1}&2&0\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&1\\0&1&1\\0&0&{ - 1}\\0&0&0\end{array}} \right)$$

From the echelon form, it can be observed that there are no free variables.

So, the polynomials are linearly independent. ### Want to see more solutions like these? 