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Find all real t such that (1+t, 1-t) and (1-t, 1+t) are independent

by RoRi

Find all values of t \in \mathbb{R} such that the two vectors (1+t,1-t) and (1-t,1+t) are linearly independent.

From the previous exercise we know that (1+t,1-t) and (1-t,1+t) are linearly independent if and only if

    \[ (1+t)(1+t) - (1-t)(1-t) \neq 0. \]

So, we have that they are independent if and only if

    \begin{align*} && (1+t)(1+t) - (1-t)(1-t) &\neq 0 \\ \implies && 1 + 2t + t^2 - 1 - t^2 & \neq 0 \\ \implies && 2t &\neq 0 \\ \implies && t &\neq 0. \end{align*}

Hence, they are independent for all t \neq 0.

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