# Linear Dependent/Spanning System?

• Jan 24th 2013, 04:00 PM
glambeth
Linear Dependent/Spanning System?
Is there a difference between a linear dependent system and a spanning one? I've been confused about this for a while. Something is linearly dependent if it has a non trivial solution, and something is spanning if one of the vectors can be represented as a linear combination of another - implying there is a non-trivial solution- Some of my classmates told me i'm wrong, and I don't understand why. An explanation would be awesome. Can't seem to find anything online.
• Jan 24th 2013, 04:34 PM
jakncoke
Re: Linear Dependent/Spanning System?
A Spanning set for a Vector space V can be linearly dependent but not every linearly dependent set is a spanning set for V.
• Jan 24th 2013, 04:38 PM
glambeth
Re: Linear Dependent/Spanning System?
How is that so?
• Jan 24th 2013, 04:44 PM
jakncoke
Re: Linear Dependent/Spanning System?
The set of linearly dependent vectors $\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ spans $\mathbb{R}^3$. Now the set of linearly dependent vectors $\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ does not span $\mathbb{R}^3$
• Jan 24th 2013, 04:50 PM
glambeth
Re: Linear Dependent/Spanning System?
A system of vectors v1; v2;...; vn in V is called a generating system (also a spanning system, or a complete system) in V if any vector v in V admits representation as a linear combination.

This is the definition from the book I am using. isn't v2 = 3*v1. Therefore it is a spanning system?
• Jan 24th 2013, 04:54 PM
jakncoke
Re: Linear Dependent/Spanning System?
Any vector in V must be admitted as a linear combination.

So the vector $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ lives in $\mathbb{R}^3$ but give me a linear combination of $\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ to produce r $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, cant be done so this linearly dependent system does not span $\mathbb{R}^3$
• Jan 24th 2013, 04:56 PM
glambeth
Re: Linear Dependent/Spanning System?
Ahhh. Thank you very much. I appreciate the help. I guess i have to pay more attention to key words!