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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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4 Answers 4

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 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 $2v = (1,1)$ 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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A span of vectors forms a linear space which means that the vectors form a basis of that space (i.e they are linearly independent and possiby orthogonal) and as a result any other vector can be writen as a unique linear combination of these basis vectors (think how cartesian coordinates work).

In general a set (span) of vectors of a linear space may not span the whole space if they are not linearly independent and/or their number is less that the dimension of the space.

When a matrix has full rank, it implies that its eigen vectors span the linear space (provide a basis for the space) and this is how linear algebra equations are solved.

Intuitively if a matrix has full rank, it has an inverse. An inverse of a function exists only when the function is 1-1 (bijection). This notion of bijection in terms of linear algebra and linear spaces is also related to full rank and vectors span

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The span of a set of vectors does not necessarily form a basis for the space (since the original vectors are not necessarily linearly independent). –  JohnD Jun 15 at 1:21
    
@JohnD True the answer states "the span of vectors of a space" and then addresses full rank matrices, will add an edit also –  Nikos M. Jun 15 at 1:24
    
The point remains: for example, if $v_1,v_2\in V$, then $\text{span}\{v_1,v_2\}$ is not necessarily a basis for $V$. Simply take $V=\mathbb{R}^2$, $v_1=e_1$, $v_2=2e_1$. Your first and second sentences contradict one another. –  JohnD Jun 15 at 1:46
    
@JohnD, i dont see how the 2 senteces contradict one another, the first says that any vector span spans a space or forms a space (which is true), the second says that a vector span of a specific space does not necesarily span the whole space (which is also true), then full rank vector spans are discussed which is the main point of the question –  Nikos M. Jun 15 at 1:53
    
@JohnD, re-phrased first sentence to match the objection –  Nikos M. Jun 15 at 1:58

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