I'm always having difficulties with what actually suffices as a proof, and what is obvious enough to not have to prove it. Here some I have those problems with.
Let $S=\{u_1,u_2,…,u_n\}$ be a finite set of vectors. Prove that $S$ is linearly dependent if and only if $u_1=0$ or $u_{k+1} \in <\{u_1,u_2,…,u_k\}>$ for some $k (1 \leq k \lt n)$.
I think it's obvious since 1. ${0}$ is linearly dependent and 2. if $u_{k+1}$ is in the mentioned span, than it is a linear combination of other vectors of $S$ and thus $S$ is linearly dependent. So much to my thinking, but how do I appropriately express something like this?
Let $M$ be a square upper triangular Matrix with nonzero diagonal entries. Prove that the columns of $M$ are linearly independent.
I think: If you regard every column as a vector, each has a different direction. But again, I guess this doesn't really count as a mathematical proof. How to express it then?
Let $V$ be a vector space over a field of characteristic not equal to two.
a) Let $u$ and $v$ be distinct vectors in V. Prove that $\{u,v\}$ is linearly independent if and only if $\{u+v,u-v\}$ is linearly independent.
( b) – the same with $\{u,v,w\}$ and $\{u+v,u+w,v+w\}$.
My proof for a):
$a_1u + a_2v$ implies $a_1=a_2=0$
If $$b_1(u+v) + b_2(u-v) = (b_1+b_2)u+(b_1-b_2)v = 0$$
then $$b_1+b_2 = b_1-b_2 = 0$$ thus $b_1 = b_2=0$
Now do I need to write additional stuff to prove the other direction (something like:
Since
$$b_1(u+v) + b_2(u-v) = (b_1+b_2)u+(b_1-b_2)v = 0$$
only if $b_1 = b_2 = 0$
I can choose no $a_1,a_2 \neq 0$ for I could always split it into $b_1$ and $b_2$ which were not zero, there fore the equation couldn't be zero either.),
or would the first part already be enough?
Thanks a lot ! I have a lot to get used to 😛
Best Answer
In the future, please limit yourself to one problem in your post. You're more likely to get helpful responses than if you list several problems.
For #1, you have given an argument for the "if" direction, but you haven't addressed the "only if" direction. You must show that if $S$ is linearly dependent, then one of the two cases stated holds.
For #2, you should take a linear combination of the columns and argue that each of the coefficients is zero. Your argument about directions is not valid. You can select three vectors in $\mathbb{R}^2$ that all have different directions. Those three vectors will be linearly dependent (why?).
For #3, your definitely need to say something for the other direction. Otherwise, you will have only proven the "only if" direction. For the "if" direction, you assume that $\{u+v, u-v\}$ are linearly independent (i.e. $b_1 (u+v) + b_2(u-v) = 0 \Rightarrow b_1 = b_2 = 0$. Then, to prove that $\{u, v\}$ is linearly independent, you assume that $a_1 u + a_2 v = 0$ and must show that $a_1 = a_2 = 0$. Your proof will be similar to the other direction, using an algebraic calculation to get equations on $a_1$ and $a_2$ whose only solution is $a_1 = a_2 = 0$.
Hope that helps. Keep practicing and asking questions - that's the best way to learn how to do proofs.