# Thread: Graphical analysis of linear systems

1. ## Graphical analysis of linear systems

Hi,

I'm in the midst of solving the question below:

I am pretty certain that b is a solution - based on the parallelogram rule (it seems to be a linear combination of c1v1 + c2v2 + 0v3). In terms of the type of solution... I'm not really sure.

One thing that I do notice, is that one of the vectors must be linearly dependent on the other since the space is R^2 and from my understanding it is impossible to have three independent vectors within the space of R^2. Whether my observation is true or not, I still am not sure if it has any relevance to answering this question.

Appreciate help.

- Olivia

2. ## Re: Graphical analysis of linear systems

Originally Posted by otownsend
I'm in the midst of solving the question below:

I am pretty certain that b is a solution - based on the parallelogram rule (it seems to be a linear combination of c1v1 + c2v2 + 0v3).
Since all of this is in $\mathbb{R}^2$ any three vectors are linearly dependent. So from your post, it seems that you do not know that. Am I mistaken?
So if we take $\{\vec{v}_1,~\vec{v}_2,~\&~\vec{b}\}$ they must be dependent.

3. ## Re: Graphical analysis of linear systems

Okay so based on what you said, b is a non-unique solution. Correct?

4. ## Re: Graphical analysis of linear systems

Originally Posted by otownsend
Okay so based on what you said, b is a non-unique solution. Correct?
Frankly, I have no idea what you could possibly mean by "b is a solution".
The given equation does have solution in terms of $\{x_1,x_2,x_3\}$ but it is not unique.

5. ## Re: Graphical analysis of linear systems

You are completely misunderstanding the problem. The question is NOT whether "b" is a solution. The question is whether there exist a triple of real numbers, $\displaystyle (x_1, x_2, x_3)$ that satisfy that equation for given b. If the exist a solution is that solution unique?

6. ## Re: Graphical analysis of linear systems

So I would say that (b) is a unique solution since it does not overlap the other vectors. If it did overlap this would imply that (b) had infinitely many solutions or if (b) wasn't in the span of x1 and x2 then (b) would not be a solution.

7. ## Re: Graphical analysis of linear systems

What in the world is "(b)"? There is no "(b)" given any where in the problem. If you are still asking whether the vector, "b", is "a solution", the answer is still "no"! The question is asking whether there exist numbers $\displaystyle x_1$, $\displaystyle x_2$, and $\displaystyle x_3$ that satisfy this equation. If there exist such a triple of numbers, that is the "solution" to the equation, not "b"..

8. ## Re: Graphical analysis of linear systems

Sorry, yes you're right!!!

What I meant to say is that a solution does exist. I believe that the solution would be unique since the vector b doesn't "overlap" with the other vectors. I could imagine that the coefficient matrix in RREF would have a zero row and no free variables. Correct?

9. ## Re: Graphical analysis of linear systems

You have also been told, by Plato in the very first response to your question, that, because this is in $\displaystyle R^2$, any vector can be written as a linear combination of two independent vectors. And each pair of $\displaystyle \{v_1, v_2\}$, $\displaystyle \{v_1, v_3\}$, and $\displaystyle \{v_2, v_3\}$ are independent because one is not a multiple of the other. b can be written as a linear combination of those three vectors in infinitely many different ways.