Question

Use the definitions for span and subspace to explain why the span of a set of vectors from V generates a subspace of V.

Span: We say the vector w from the vector space V is a linear combination of the vectors v1, v2, ..., vn,
all from V, if there are scalers c1, c2, ... , cn so that w can be written w = c1v1+c2v2+ ... cnvn

Subspace: Suppose that V is a vector space and W is a subset of V. If, under the addition and scalar multiplication that is defined on V, W is also a vector space then we call W a subspace of V.

I know this should be easy but I'm having a hard time figuring it out x_x.

Answer 1

where are you stuck? You should have a little theorem that will help you?

Answer 2

im stuck on the spot where its asking to explain why and all thats given are these definitions

Answer 3

oh..i guess I should read all that you posted...the second part you have there is not a def but the theorem that I was thinking of

Answer 4

in the book i have its the definition XD

Answer 5

by the way in the definition for span the numbers and the n is sub

Answer 6

pick two vectors that are in the span (uv) and a constant (c)...

show that u+v is in the span and cv is in the span

Answer 7

Let \[S=span\{v_1,v_2,\ldots v_k\}\]
let \[u,w\in S\]

then \[u=c_1v_1+\cdots +c_kv_k\]
\[w=b_1v_1+\cdots +b_kv_k\]

\[u+w=(c_1+b_1)v_1+\cdots +(c_k+b_k)v_k\]

which is a linear combination of the vectors of S. thus u+w is in the span of S

do the same for scalar mult

Answer 8

Im not entirely sure how that explains why the span of a set of vectors from V generates a subspace of V though

Answer 9

Theorem 1 Suppose that W is a non-empty (i.e. at least one element in it) subset of the vector space V then W will be a subspace if the following two conditions are true.

(a) If u and v are in W then u+v is also in W (i.e. W is closed under addition).

(b) If u is in W and c is any scalar then cu is also in W (i.e. W is closed under scalar multiplication).

Answer 10

if you have to you can go through all the properties of a vector space and show that the span satisfies them all.

Answer 11

so then you could say by the theorem you posted that a set of vectors from V generates a subspace because:
(a) If u and v are in W then u+v is also in W (i.e. W is closed under addition).

and

(b) If u is in W and c is any scalar then cu is also in W (i.e. W is closed under scalar multiplication).?

Answer 12

yes

Answer 13

if you can show that a subset satisfies (a) and (b) then it is a subspace

Answer 14

noiiice :D totally understand that XD tyvm Zarkon

Answer 15

np