Question

let , be two vectors in a vector space V. Show that , are linearly dependent if and only if one of the vectors is a scalar multiple of the other.

Answer 1

I understand that it is clear the two vectors are linearly dependent, I'm just having trouble proving this algebraically.

Answer 2

Why do it algebraically if you can plot them in MATLAB :-D

Answer 3

oh.... can't be visualized using MATLAB :(

Answer 4

You can prove that two vectors are linearly dependent by computing their Wronskian.

Answer 5

... or equivalent thereof.

Answer 6

really? Had no idea the wronskian was used aside from variation of parameters.

Answer 7

something about it being 0 and whatnot

Answer 8

It's much easier than that, isn't it? Two vectors v and w are linearly dependent, by definition if there exist _non-zero_ constants a and b such that
av + bw = 0.

Now from that, you should be able to prove very easily your result.

Answer 9

That's what I had originally thought. I was then able to say that (a+kb) = 0. This doesn't prove anything, but I'm thinking it might be a basis for the nullspace?

Answer 10

From there I figured I could say that the existence of the nullspace proves linear depence.

Answer 11

Let\[\vec{v}=\begin{pmatrix}
v_1\\
v_2
\end{pmatrix},\]\[\vec{u}=\begin{pmatrix}
u_1\\
u_2
\end{pmatrix},\]\[\vec{u}=c\vec{v},\exists c\in\mathbb{R}\setminus\{0\}.\]Then\[\begin{vmatrix}
v_1 & u_1\\
v_2 & u_2
\end{vmatrix}=\begin{vmatrix}
v_1 & cv_1\\
v_2 & cv_2
\end{vmatrix}=...\]

Answer 12

james is right. dont make this problem way harder than it has to be >.<

Answer 13

If v and w are Lin Dep, there exists nonzero constants a and b such that
av + bw = 0 => v = (-b/a)w. Thus one is scalar multiple of the other.

Conversely, suppose wlog that v = kw for some constant k. Then v + (-k)w = 0. Hence v and w and Lin Dep.

Answer 14

I understand the first proof provided. It makes perfect sense. However, the second one makes less sense. I don't understand how changing the constant to some other arbitrary constant proves linear dependence.

Answer 15

Let v and w be two vectors in a vector space V. Then let A be the statement: "v and w are linearly dependent" and let B be the statement "v can be written as a scalar multiple of w or vice versa"

We want to show that A => B and B => A

The proof of A => B is the first thing:
If v and w are Lin Dep, there exists nonzero constants a and b such that
av + bw = 0 => v = (-b/a)w. Thus one is scalar multiple of the other.

The proof of the converse, B => A is now this:
Suppose wlog (without loss of generality) that v = kw for some constant k. Then v + (-k)w = 0. Hence v and w and Lin Dep.

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We write v = kw for some constant k because that's our hypothesis; we're starting with the statement B; this mathematical statement is what we know.

Answer 16

OH! Right. Hah, that's quite obvious now. Thanks!