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I am having trouble understanding the span of functions, my problem is:

What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension?

I understand this in terms of vectors, but not in terms of functions, so any enlightenment would be very much appreciated.

Ellya
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1 Answers1

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Enlightenment: Functions are vectors.

Assuming you mean $a$ and $b$ are fixed constants in your base field $F$ (probably $\mathbb{R}$ or $\mathbb{C}$), then $$ \text{Span}_F\{a\sin x,b\} = \{f_1 a\sin x + f_2 b : f_1,f_2 \in F\}. $$

When viewing $v = a \sin x$ and $w = b$ as vectors, the equation $$ f_1 a\sin x + f_2 b = f_1 v + f_2 w = 0 $$ means that for all values of $x \in F$ (or whatever the domain of $\sin x$ is assumed to be) $$ f_1 a\sin x + f_2 b = 0. $$ Are there any constants $f_1$ and $f_2$ that make this true? If not, then the span has dimension $2$. If so, its dimension is less than two, since this says $v$ and $w$ are linearly dependent.

Dan
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  • To make it true you would need $f_1,f_2\ne 0$ s.t $f_1a\sin(x)+f_2b=0,,\forall x\in[0,1]$? (my function domian is $[0,1]$), in which case this is not true so I would believe the dimension is 2 correct? – Ellya Apr 07 '14 at 19:48
  • That's correct! The dimension is in fact $2$, and for precisely the reason you mention. – Dan Apr 07 '14 at 20:15