- #1

karnten07

- 213

- 0

**[SOLVED] Spans and linear independance**

## Homework Statement

Let V be a vector over a field F.

a.) Let x1,...,xn[tex]\in[/tex]V and y1,...,ym[tex]\in[/tex]V. Show that

Span(x1,...,xn,y1,...,ym) = Span(x1,...,xn) + Span(y1,...,ym)

B.) Let x1, x2, x3, x4 be four linearly independent vectors in V. Show hat

Span(x1, x2,x3) [tex]\cap[/tex] Span(x2, x3, x4) = Span(x2,x3)

c.) Show that the equality in part b.) does not hold if we drop the assumption that x1, x2, x3, x4 are linearly independent.

## The Attempt at a Solution

a.) Does it suffice to show:

For a,b[tex]\in[/tex]R,

(a1x1+...+anxn+b1y1+...+bmym) = (a1x1+...+anxn)+(b1y1+...+bmym) ?

b.)

Does it suffice to show:

For a,b[tex]\in[/tex]R,

Span(x1, x2,x3) [tex]\cap[/tex] Span(x2, x3, x4)= (a1x2(1)+...+anx2n+b1x3(1)+...+bmx3m) = Span(x2, x3) ?

c.) If what i have doen so far is reasonably correct, the only part I'm unsure about is part c.) I will do some reading about it if i can, but any hints would be greatly appreciated. Thanks