Geometrically, what is the span of vectors?

Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?

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In $R^n$:

If two vectors are linearly dependent their span is the line determined by the vectors (the line made by a vector starting at the origin).

If two vectors are linearly independent their span is the plane.

For three linearly independent vectors the span is the entire three dimensional space.

If the three vectors are linearly dependent then it is either a plane or a line depending on "how linearly dependent" the vectors are.

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i found this to be very enlightening. – J L Nov 9 '14 at 4:34

Well, the span of a single vector is all scalar multiples of it. For example, if you have $\mathbf{v} = (1,1)$, $\text{span}(v)$ is all multiples of $(1,1)$. So $2v = (2,2)$ is in the span, $-3.75v = (-3.75,-3.75)$ is in the span, and so on. What you end up with is the whole line $y=x$, which is what you get if you extend $v$ infinitely in either direction. Note that this is determined by it's slope. So the span of two vectors with the same slope is still just the same line.

Now, the span of two vectors are all of the combinations $a\mathbf{v} + b\mathbf{w}$. So if you have $\mathbf{v} = (1,1,0), \mathbf{w} = (0,1,0)$, you can get $$(3,5,0) = 3\mathbf{v} + 2\mathbf{w},\quad (7.14,-3.86,0) = 7.14\mathbf{v} - 11\mathbf{w},$$ and in fact any other vector in the plane determined by the 3 points $(0,0,0), (1,1,0), (0, 1,0)$ (since 3 points determine a plane). In this case, you get the $xy$-plane.

The span of more than two vectors is defined similarly.

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Assuming it makes sense that the span of a single vector is a line, we can imagine the two vectors in 3-space. Because the span of each vector lies within the space of each of them, we can draw the two lines that are in the direction of these two vectors:

• if the two lines are equal, then this is all of the span.

• otherwise, we can imagine picking one of the lines and sliding it along the other line such that it stays parallel to its original placement.

To make the above intuition precise, the point is that the span of a single vector is the result of performing scalar multiplication on that single vector: that is, contracting it, shrinking it, or even flipping its orientation (making the direction go the other way). Then we can get any other point of the plane by finding the projection of the point onto the first line, and then translating by the projection of the point onto the second line; this is why when the lines are the same we do not leave the line.

As far as the formal definition of the span goes, the span of a set $S=\{v_1,\ldots, v_n\}$ of vectors is given by the set $$\mathrm{span}(S)=\left\{ \sum_{i=1}^n{c_iv_i} \mid c_i\in \mathbb{F}, v_i\in S\right\}$$ where $\mathbb{F}$ is the field that you're working over (likely the real numbers $\mathbb{R}$).

In the case where $S=\{v_1,v_2\}$, we're looking at the set of vectors of the form $c_1v_1+c_2v_2$. Now, if $v_1$ and $v_2$ are in the same direction, then there is $c$ such that $v_1=cv_2$, so we can rewrite this as $c_1v_1+c_2v_2=c_1v_1+c_2cv_1=(c_1+c_2c)v_1$, which is just any element in the span of $\{v_1\}$. This is why it is possible for the span to be a line.

Now, when they aren't in the same direction, they lie precisely in a unique plane. Any point in the line that goes along the direction of the vector $v_1$ can be written as $c_1v_1$, and any point in the line that goes along the direction of the vector $v_2$ can be written as $c_2v_2$. Every point in the plane will then be of the form $c_1v_1+c_2v_2$ by projecting the point onto $v_1$ and $v_2$ to find $c_1$ and $c_2$, respectively.

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Here is a visual way of thinking about this (at least in small enough dimension that you can visualize it).

Say you have some collection of vectors and you place yourself at the origin. Each vector determines a direction (the span does not care if you replace a vector by some non-zero scalar multiple), and the span is everywhere you are able to go by only going in the specified directions (though you are allowed to go both forwards and backwards).

So, if you were in $3$ dimensions and had the vectors $(1,0,0)$ and $(0,1,0)$ then you could for example go to $(-1,2,0)$ by first going backwards $1$ unit in the direction of the first vector and then going forwards $2$ units in the direction of the second vector.

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