The vectors

v =

are linearly independent if and only if k does not equal ______?

Ok so i set up the matrix

then i reduced it down to this

Not sure what to do now or if i am even headed in the right direction :(

Thanks for any suggestions.

Printable View

- Mar 28th 2010, 12:58 PMmybrohshi5[SOLVED] Vectors - Linear independence
The vectors

v =

**are linearly independent if and only if k does not equal ______?**

Ok so i set up the matrix

then i reduced it down to this

Not sure what to do now or if i am even headed in the right direction :(

Thanks for any suggestions. - Mar 28th 2010, 05:16 PMvincent
you're almost there! u, v and w are linearly independant when the equation only admits the trivial solution which is when the scalars . you need to find a k such that your system of equations satisfies this condition. this is the same as saying that your matrix can be reduced to a diagonal matrix!

- Mar 28th 2010, 06:18 PMBruno J.
A perhaps shorter way is to evaluate the determinant of the matrix whose columns (or rows) are your three vectors, and find the values of for which this determinant does not vanish. Make sure you understand why this works, however - otherwise it's cheating! (Happy)

- Mar 28th 2010, 08:14 PMmybrohshi5
- Mar 28th 2010, 08:29 PMBruno J.
No! First off, you should avoid dividing by an expression containing because the expression might be zero. In any case, I believe you also made a calculation mistake (the numerator of the fraction should be and not ).

When you reach

you can subtract times the second row from the third, to get

No divisions involved!

Now you want the bottom right corner to be nonzero, i.e. you want .

Check to see if you get the same answer with the determinant method. I didn't check all your calculations so I can't guarantee you that your answer is correct! (Nod) - Mar 28th 2010, 08:35 PMmybrohshi5
Thank you. 13 was correct.

I will do it using determinants and see if i get the same answer :)

Thanks again for all the help