(adsbygoogle = window.adsbygoogle || []).push({}); LINEAR ALGEBRA: 3 vecotrs in R^4 (with 6 variables) -- Are they linearly independent?

For which values of the constants a, b, c, d, e, anf f are the following vectors linearly independent? Justify your answer.

[tex]\left[\begin{array}{c}a\\0\\0\\0\end{array}\right],\,\,\left[\begin{array}{c}b\\c\\0\\0\end{array}\right],\,\,\left[\begin{array}{c}d\\e\\f\\0\end{array}\right][/tex]

I figure that one would setup an equation:

[tex]x\,\left[\begin{array}{c}a\\0\\0\\0\end{array}\right]\,\,+\,\,y\,\left[\begin{array}{c}b\\c\\0\\0\end{array}\right]\,\,+\,\,z\,\left[\begin{array}{c}d\\e\\f\\0\end{array}\right]\,\,=\,\,\left[\begin{array}{c}0\\0\\0\\0\end{array}\right][/tex]

[tex]x\,a\,\,+\,\,y\,b\,\,+\,\,z\,d\,\,=\,\,0[/tex]

[tex]\,\,\,\,\,\,\,y\,c\,\,+\,\,x\,e\,\,=\,\,0[/tex]

[tex]\,\,\,\,\,\,\,\,\,\,\,\,\,\,z\,f\,\,=\,\,0[/tex]

How does one proceed?

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# LINEAR ALGEBRA: 3 vecotrs in R^4 (with 6 variables) - Are they linearly independent?

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