how would i prove that for a vector space V, a subset of V that contains the zero vector is linearly dependent?
Isn't any vector space containing the zero vector linearly dependent or am I misreading this?
Consider the nonempty set $\displaystyle S = \{\bold{v}_{1}, \bold{v}_{2}, \hdots, \bold{v}_{n}, \bold{0}\}$. From here, we can see that: $\displaystyle 0\bold{v}_{1} + 0\bold{v}_{2} + \hdots + 0\bold{v}_{n} + 1(\bold{0}) = \bold{0}$
i.e. 0 is a linear combination of the vectors in S whose coefficients aren't all 0.
I have a problem with that definition of linear dependence and this is the ideal thread to get some opinions.
The definition of linear dependent I like to use is that any vector S can be constructed as a linear combination of the others.
Consider the set S = {0, i, j). Linearly dependent or not? I say not since you can't construct i as a linear combination of 0 and j .....
(I do understand the argument when the other definition is used that S is dependent ..... I just don't like it ......)
Hmmm..... What about $\displaystyle S = \{(1,2),(2,4),(1,1)\}$?
This set is clearly linearly dependent. But I cant express each element of S as a linear combination of the others. For example, (1,1) cannot be expressed as a linear combination of the others, yet S is linearly dependent.
Considering your knowledge, perhaps you meant:
Technically this statement is wrong too, unless you define it that way. A set S is defined to be linearly dependent if it is not linearly independent. So technically(formally, logically...) you have to negate the definition of linear independence to get the mathematical formulation for linear dependence. And that formulation is the one you dont like....Originally Posted by Mr.F meant
I think this dilemma of implication and equivalence is pretty common. Generally if you have a set of vectors S where one is able to express some element of S as a linear combination of the others, then the set is definitely linearly dependent. However the other way round is not necessarily true. And your question works as wonderful illustration to this fact.
These are my thoughts and it could be unconvincing. Perhaps some MHF Algebraist can explain it better.
Thanks for this reply. In addition:
Linear Dependence Theorem: The set {$\displaystyle v_1, \, v_2, \, .... v_n$} of non-zero vectors is linearly dependent if and only if some $\displaystyle x_k$, $\displaystyle 2 \leq k \leq n$, is a linear combination of the preceding ones.
Theorem: A set of two or more vectors $\displaystyle S = ${$\displaystyle v_1, \, v_2, \, .... v_n$} is linearly dependent if and only if one of the $\displaystyle v_i$ is a linear combination of the other vectors in S.