Question

Determine the conditions on the scalars so that the set of vectors is linearly dependent.
v1=[1
a]
v2=[2
3]

Answer 1

two vectors are dependent if one is a constant multiple of the other

Answer 2

i know that, but i'm trying to find a more general way to solve this problem because i have other problems that include multiple scalars/variables

Answer 3

What Zarkon said is the general condition.

Writing your vectors in row vs. column form, v1 and v2 (both of which are not the zero vector) are linearly dependent if and only if there is a scalar k such that:

v1 = k.v2 or v2 = k.v1

Let's use the first form

(1,a) = k(2,3) = (2k,3k)
iff
1 = 2k and a = 3k
iff
k = 1/2 and a = 3k = 3/2

I.e., the vectors are linearly dep. iff a = 3/2

Answer 4

that is fine for that problem, but i'm not seeing how i could use that method to solve something like v1=(1, 2, 1) v2=(1,a,3) v3=(0,2,b)

Answer 5

make a matrix and row reduce it or find the determinant.

Answer 6

i did that and ended up getting the answer, but i dont know why i did one of the steps, and it's kind of hard to write it all in matrix form on here

Answer 7

Right. Because the row vectors of a nxn matrix A are
lin indep <=> det(A) is not zero <=> the rank of A = n <=> there is no row reduction with a zero row

Answer 8

that is fine for that problem, but i'm not seeing how i could use that method to solve something like v1=(1, 2, 1) v2=(1,a,3) v3=(0,2,b)

Answer 9

so i got to [1 1 0 0
0 1 -2/2-a 0
0 0 -4-b 0]

i then set -2/2-a equal to -4-b and solved, but why is that correct?

Answer 10

Because they are equal, then you can write down a row reduction with a zero row. Or alternatively, the matrix will have zero determinant, however you want to look at it.

Or in other words, you are able to write one of the vectors as a linear combination of the two others.

Answer 11

if you have that zero row, doesn't that make x2=0, thus making x1=0 as well which means it is linearly independent? sorry im just not grasping this