Question

If we let W={u,v1,v2,...,vk} and suppose that u is a linear combination of the vectors in the set S={v1,v2,...,vk}, i.e W with u removed. Prove that spanW=spanS.

I understand the concept visually but how would you write the proof?

Answer 1

$$\DeclareMathOperator{\span}{span}$$consider by definition every vector in \(w\in\span W\) may be written in the form:$$w=\lambda_{k+1} u+\sum_{i=1}^k\lambda_i v_i$$if \(u\) is a linear combination of vectors in \(S\) we write:$$u=\sum_{i=1}^k\mu v_i$$hence \(w\) is written:$$w=\lambda_{k+1}\sum_{i=1}^k\mu v_i+\sum_{i=1}^k\lambda_i v_i=\sum_{i=1}^k(\lambda_{k+1}\mu+\lambda_i)v_i$$and therefore \(w\in\span S\) so it follows \(\span W\subseteq\span S\). Now prove \(\span S\subseteq\span W\) and we arrive at \(\span W=\span S\) by antisymmetry.

Answer 2

ops those should be \(\mu_i\) above

Answer 3

ah right. we did something similar in our lectures. so we do the same thing with w∈spanW? and i'm guessing anti-symmetry just illustrates that if both ways come to the same conclusion then spanW=spanS?

Answer 4

e.g. every vector \(s\in\span S\) may be written \(s=\sum\limits_{i=1}^k\lambda_i v_i=\sum\limits_{i=1}^k(\mu_i+\lambda_i-\mu_i)v_i=\sum\limits_{i=1}^k\mu_i v_i+\sum\limits_{i=1}^k(\lambda_i-\mu_i)v_i=u+\sum\limits_{i=1}^k(\lambda_i-\mu_i)v_i\) therefore \(s\in\span W\) therefore \(\span S\subseteq\span W\) and \(\span W\subseteq\span S\land\span S\subseteq\span W\Longleftrightarrow\span W=\span S\) (anti-symmetry of \(\subseteq\))

Answer 5

exactly @chris00

Answer 6

what does this ∧ symbol mean?

Answer 7

logical and

Answer 8

why is that necessary? i'm just familiar with that notation yet.

Answer 9

well I'm just stating that if \(\span W\subseteq\span S\) and \(\span S\subseteq\span W\) we can infer \(\span W=\span S\) by definition

Answer 10

ah too true. i'm amazed at your maths abilities oldrin. keep up the good work!

Answer 11

that is very familiar with what you have written. cheers!

Answer 12

haha thank you and I will try

Answer 13

hey oldrin, i'm just wondering what is \[\mu_i\] for?

Answer 14

@oldrin.bataku

Answer 15

@chris00 just variables to denote the coefficients

Answer 16

is that the same for lamda?

Answer 17

is that a variable as well?

Answer 18

each is a variable

Answer 19

i'm just not seeing the connection between the two

Answer 20

http://en.wikipedia.org/wiki/Linear_span#Definition

Answer 21

thanks for that. also, do you know why u used \[\lambda _{k+1}\]? for u?

Answer 22

@oldrin.bataku

Answer 23

@chris00 well I used \(\lambda_0,\dots,\lambda_k\) for the coordinates w.r.t. \(v_0,\dots,v_k\), but since our first basis uses \(u\) I needed another coordinate (namely \(\lambda_{k+1}\)) :-p