Difference between vectors and span Let $A = 
    \begin{bmatrix}
     1       & 0 & -4 \\
     0       & 3 & -2 \\
     -2       & 6 & 3 
    \end{bmatrix}
$ and $b=\begin{bmatrix}
     4   \\
     1    \\
     4   
    \end{bmatrix}$.
Denote the columns of $A$ by $a_1,a_2,a_3$ and let $W=Span\{a_1,a_2,a_3\}$ 
How many vectors are in $\{a_1, a_2, a_3\}$? Correct answer: 3
How many vectors are in $W$? Correct answer: infinitely many
I'm confused on the difference between the two answers, and why the answer to both isn't infinitely many. I thought that the vector $a_1$ and $2*a_1$ are equivalently the same. 
 A: *

*When we write $\{a_1,a_2,a_3\}$, it's simply a set notation denoting a list consisting in $a_1,a_2$ and $a_3$.  In this notation, we have $\{a,a,b\} = \{a,b\}$.  Since $a_1,a_2$ and $a_3$ are distinct, $\{a_1,a_2,a_3\}$ has three vectors.  This has nothing to do with linear algebra.

*$W=\mathrm{span}\{a_1,a_2,a_3\} = \{c_1a_1 + c_2a_2 + c_3a_3 \mid c_1,c_2,c_3 \in \Bbb F\}$, where $\Bbb F$ is the field (usually $\Bbb R$ or $\Bbb C$) on which the vector space is defined.  So $W$ is a linear combination of $a_1,a_2$ and $a_3$, so $W$ has infinitely any elements.

A: Two vectors in $\mathbf{R}^{3}$ are equal if and only if their corresponding components are equal. If $a \neq 0$, then $a$ and $2a$ (for example) are distinct. Since the columns of the matrix $A$ are distinct, the set $\{a_{1}, a_{2}, a_{3}\}$ has three elements. If $A$ had been the zero matrix, or (for example) the matrix
$$
A = \left[\begin{array}{ccc}
1 & 1 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right],
$$
its set of columns would have had just one element.
A: The span of a set of vectors $\{a_1, a_2, a_3\}$ is the set of all linear combinations of those vectors. For example $3a_1+2a_2-a_3$ is in $W$, because it can be written as the linear combination above. So, the span of this set has infinitely many vectors in it, because there are infinitely many scalars to choose from in this case. However, the spanning set may have only finitely many vectors. Also note that $a_1 \not= 2a_1$ because the vectors do not contain the same components. Think of the set of vectors like a recipe that only tells you what ingredients to use, but not how much of each to use: you can make infinitely many dishes by adding a little more, or a little less of each ingredient, but you only have finitely many ingredients.
A: $\{a_1, a_2, a_3\}$ is simply the set containing the three vectors, hence the answer: $3$.
The column vectors $a_1, a_2, a_3$ are linearly independent. You can verify this by checking that the determinant is non-zero.
$W$, the span of $\{a_1, a_2, a_3\}$ is the set of all linear combinations of the three vectors, and hence contains infinitely many vectors(provided that at least one of the three vectors is non zero). In addition, if the three vectors are linearly independent, as in this case then $W$ will be all of 3D space i.e. all 3D vectors($\mathbf{R}^3$).
If you have the time and are so inclined, I suggest you at least read through the proofs of elementary Linear algebra. You might quite enjoy them, as I did(sometimes you can almost 'see' 4D vectors in your head ;) ).
