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.

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.

Re: Linear Dependent/Spanning System?

Re: Linear Dependent/Spanning System?

The set of linearly dependent vectors $\displaystyle \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ spans $\displaystyle \mathbb{R}^3 $. Now the set of linearly dependent vectors $\displaystyle \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ does not span $\displaystyle \mathbb{R}^3$

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?

Re: Linear Dependent/Spanning System?

**Any** vector in V must be admitted as a linear combination.

So the vector $\displaystyle \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} $ lives in $\displaystyle \mathbb{R}^3 $ but give me a linear combination of $\displaystyle \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \}$ to produce r $\displaystyle \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} $, cant be done so this linearly dependent system does not span $\displaystyle \mathbb{R}^3$

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!