Relationship between determinants and basis formation

[neden]

Hi,

I'm scratching my head over the statement from my textbook which states when determinant is non-zero, the set of vectors blah blah is a basis for r^3.

That does not make any sense to me because I know when a row of zeros in a matrix occur; the determinant is zero (through Gaussian Elimination), which means that a column has to be parameterized; then you are to set a arbitrary variables accordingly and from that one may create the needed basis.

What am I not understanding?

 

[HallsofIvy]

I'm afraid what you are saying makes no sense to me. I assume that the determinant you are talking about is the determinant of a matrix having the given vectors as columns. Second, I assume you are talking about a set of three given vectors in R³.

Yes, it is true a set of three vectors in R³ will form a basis for R³ if and only if the determinant of the matrix having those vectors as columns (or rows) is non-zero.

That is true because three vectors will be a basis for R³ if and only if they are independent: that the equation
$$x\begin{bmatrix}a \\ b\\ c\end{bmatrix}+ y\begin{bmatrix}d \\ e\\ f\end{bmatrix}+ z\begin{bmatrix}g \\ h \\ i\end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}$$
has only the "trivial" solution x= y= z= 0.

That is the same as saying that the three equations ax+ dy+ gz= 0, bx+ ey+ hz= 0, and cx+ fy+ iz= 0 have only the solution x= y= z= 0.

And that is true if and only if the "matrix of coefficients"
$$\begin{bmatrix}a & d & g \\ b & e & h \\ c & f & i\end{bmatrix}$$
has non-zero determinant.

But I have no idea what you mean by "which means that a column has to be parameterized". What do you mean by "parameterizing" a column?

 

[Edgardo]

Have a look at the http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lindep/lindep.html" . Three vectors u,v and w are tested for linear dependence.
The corresponding matrix has a row of zeroes (after row reduction). The parameter is then z and you can show that the vector w can be expressed as a linear combination of two other vectors u and v.