Question

For which values of k are the vectors u = (1, k, 2), v = (-3,4, I), w = (k, 0,l)
linearly independent?

Answer 1

what does linearly independant mean?

Answer 2

it has something to do with column vectors and a determinant of zero

Answer 3

linearly independent means the sum of the vectors does *not* equal zero

Answer 4

\begin{array}c
u&v&w\\
1&-3&k\\
k&4&0\\
2&l&l\\
\end{array}

1(4.l - l.0) + -3(0.2 - k.l) + k(k.l - 2.4) = N ; if N = 0 they are dependant

Answer 5

the rest of it id have to reread alot of stuff to remember :)

Answer 6

cheers bbz

Answer 7

its weird that you have 2 variables floating around in there =/ another way to show linear independence is to row reduce the matrix to the identity matrix. thats a little rough with the two variables though.

Answer 8

i wonder if we can trial and error it :)

Answer 9

joemath, the l's are meant to be number one's

Answer 10

oh <.< then its not that hard. let me see if i can do it on paper, one sec lol

Answer 11

.... lol, you know how many synapses i just blew out trying to solve for l and k ...

Answer 12

and amistre's method might be more efficient. you find the det of the matrix in terms of k, and you dont want it to be zero. so find out what values make it 0, and just say k can be anything but those values.

Answer 13

\[1(4.1 - 1.0) + -3(0.2 - k.1) + k(k.1 - 2.4) = N\]
\[4 + 3k + k^2 - 8k = N\]
\[k^2 - 5k+4 = N\]

Answer 14

just from observation, k = 4 is a bad value.

Answer 15

thank you all.

Answer 16

f(k) = 0 when k = 4 or 1