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Expert-verified Found in: Page 1 ### 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 # Determine the value(s) of $$a$$ such that \left\{ {\left( {\begin{aligned}{*{20}{c}}1\\a\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}a\\{a + 2}\end{aligned}} \right)} \right\} is linearly independent.

The vectors are linearly independent for all values of $$a$$, except $$a = 2$$ and $$a = - 1$$.

See the step by step solution

## Step 1: Write the vector in the augmented matrix form

Write the vector in the augmented matrix form.

\begin{aligned}{l}{x_1}{{\mathop{\rm v}\nolimits} _1} + {x_2}{{\mathop{\rm v}\nolimits} _2} = {{\mathop{\rm v}\nolimits} _3}\\{x_1}\left( {\begin{aligned}{*{20}{c}}1\\a\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}a\\{a + 2}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\end{aligned}} \right)\,....\left( * \right)\\\left( {\begin{aligned}{*{20}{c}}1&a&0\\a&{a + 2}&0\end{aligned}} \right)\end{aligned}

## Step 2: Apply the row operation

At row two, multiply row one by $$a$$ and subtract it from row two.

\begin{aligned}{l}\left( {\begin{aligned}{*{20}{c}}1&a&0\\0&{a + 2 - {a^2}}&0\end{aligned}} \right)\\\left( {\begin{aligned}{*{20}{c}}1&a&0\\0&{\left( {2 - a} \right)\left( {1 + a} \right)}&0\end{aligned}} \right)\end{aligned}

## Step 3: Determine the value of $$a$$

The columns of matrix $$A$$ are linearly independent if and only if the equation $$Ax = 0$$ has only a trivial solution.

There is a non-trivial solution for equation (*) if and only if $$\left( {2 - a} \right)\left( {1 + a} \right) = 0$$.

Thus, the vectors are linearly independent for all values of $$a$$, except $$a = 2$$ and $$a = - 1$$. ### Want to see more solutions like these? 