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

1. Oct 9, 2006

VinnyCee

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.

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

I figure that one would setup an equation:

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

$$x\,a\,\,+\,\,y\,b\,\,+\,\,z\,d\,\,=\,\,0$$
$$\,\,\,\,\,\,\,y\,c\,\,+\,\,x\,e\,\,=\,\,0$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,z\,f\,\,=\,\,0$$

How does one proceed?

2. Oct 10, 2006

Office_Shredder

Staff Emeritus
You want to find a rule for when x, y, and z are all zero. Start with zf=0. When is z not zero?

Then you can take a look at your second equation, which should be yc + ze ;)

3. Oct 10, 2006