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I was trying to understand spans and it got me even more confused.

My notes about span came about like this:

Let $ S = \{ u_1, u_2... u_k \} $ be a set of vectors in $ \mathbb R^n$.

The set of all linear combination of $ u_1, u_2, .. u_k $

$$ \{ c_1u_1 + c_2u_2 + ... + c_ku_k | c_1, c_2, ... c_k \} $$

is called a linear span of S (or the linear span of $ u_1, u_2, ... u_k $) and is denoted by $ span(S) $ (or $ span( u_1, u_2, ... , u_k) $

So what is the difference when $ S = \{u_1, u_2... u_k\} $ vs $ span( u_1, u_2... u_k) $ or $span(S)$ in terms of linear combinations - Isnt $S$ also a set of all linear combinations?

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No, the notation $\{u_1,\cdots,u_n\}$ means just those $n$ vectors that you see named there. It’s a set with $n$ elements. On the other hand, span$(u_1,\cdots,u_n)$ means all possible linear combinations of those. Thus span$(u_1,\cdots,u_n)$ is an infinite set.

Best to look at simple examples: in three-space $\Bbb R^3$, consider $u_1=(1,2,0)$ and $u_2=(3,2,0)$. Then $\{u_1,u_2\}$ has two vectors in it, while the corresponding span is, and I hope you see this immediately, the whole $xy$-plane, given by the equation $z=0$.

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The span is always on a set of vectors and not on ‘a single’ vector. By the end of this response, it will be clear why span is defined on a set of vectors and not on a single vector.

Let me try to explain with an analogy from simple mathematics. Addition of 2 numbers can modelled with an equation ‘A + B = C’. where A,B,C can be any number between ‘-infinity to +infinity’(including zero). Lets try calculate addition for various values of A & B and capture the result in a table as shown below.

Span of 2 numbers A,B

In the above table, all possible values of C(highlighted in yellow colour) is the span for all possible values of numbers A, B. We can define ‘Span’ for the addition of 2 numbers as all possible values of C that can be obtained for all possible values of A + B where A,B,C can be any values ranging between -infinity and +infinity.

Lets extend the above analogy for Vectors.

You are buying cheese and butter from you local super market. Cheese costs and weighs 5 dollars and 2 pounds per box. Butter costs and weighs 8 dollars and 3 pounds per box. You can buy any number of cheese and butter boxes. As an individual you might buy lesser quantities whereas if you are purchasing for a restaurant you might buy in large quantities.

The Vector equations for the above scenario are as follows Cheese Vector = ( 5$ per box, 2 lbs per box) = C( 5, 2) Butter Vector = ( 8$ per box, 3 lbs per box) = B(8, 3) Resultant Vector( when you buy both) = Cheese Vector + Butter Vector = ( price of Cheese per box + price of Butter per box , Weight of Cheese = Weight of Butter) = R ( 5$ + 8 $, 2lbs + 3 lbs) = R(13, 5)

So the vector equations are C(5,2) + B(8,3) = R(13,5). Where C,B,R are the coefficients or scalars for the number of boxes purchased.

Depending your lucky day(Black Friday or special promotion etc.), you might get one as free if you buy the other one. So B or C can be represented as negative Lets try calculate addition for various values of C & B and capture the result in a table as shown below. Span of butter and cheese vectors

In the above table, all possible values of Resultant Vector-R (highlighted in yellow colour) is the span of Cheese and Butter vectors.

Since we are interested only in 2 dimensions(price and weight), the span(possible values of R) is the entire 2-D plane where price and weight form each of the dimensions or axes. i.e. R vector can be calculated or added or placed as a coordinate points with in the 2D plane.

If we were interested in 3 dimensions(price, weight, tax) then the span(possible values of R) is the entire 3-D plane where price, weight and taxes form each of the dimensions or axes.

For B,C we can very well have decimal values as well if your local super market allows i.e. buy one and half box of cheese, two and half box of butter etc. That's all to it!!!

So the ‘Span’ for buying of cheese and butter(read as Vector space of cheese and butter vectors) is all the possible values of R that can be obtained for all possible values of C, B where B,C can be any values ranging between -infinity and +infinity. As it is evident, span is on both cheese and butter vectors and not on only one. if it is only one, the usual number line arithmetic itself is sufficient. Hope this is clear.

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