$Linearly independent vector subsets.$

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For which t ? Q is the vectors $(t, 0, 2)^T $ , $(0,t, 1)^T $ ,$(2, 2t, t)^T $ existing subset of Q^3×1 linearly independent? And how do I prove it?

Algebra Vector Spaces Linear Algebra

Annaaa

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Mathe

Very low bounty. Also, is the T meant to mean transpose?
Annaaa

yes

Mathe

Very low bounty. Also, is the T meant to mean transpose?

Answer 1

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The three vectors are independent if
$$det \begin{pmatrix} t& 0 & 2 \\ 0 & t & 1 \\ 2 & 2t & t \end{pmatrix} \neq 0.$$
Thus
\[t \begin{vmatrix} t & 1 \\ 2t & t \end{vmatrix}+2\begin{vmatrix} 0 & t \\ 2 & 2t \end{vmatrix} \neq 0 \]
\[\Rightarrow t(t^2-2t)+2 (0-2t) \neq 0\]
\[\Rightarrow t(t^2-2t-4) \neq 0\]
\[\Rightarrow t \neq 0 \text{and} t \neq \frac{2\pm\sqrt{4+16}}{2}=\frac{2\pm\sqrt{20}}{2}=1\pm \sqrt{5}.\]
Hence if $ t \neq 0$ and $t \neq 1\pm \sqrt{5} $, then the given three vectors are linearly independent.

Nirenberg

$Linearly independent vector subsets.$

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