Let V be a vector over a field F.
a.) Let x1,...,xn∈V and y1,...,ym∈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) ∩ 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
I have done a and b just not sure about c:
If x1=x2=x3=x4 and For a ∈ in R,
Span(x1, x2,x3) ∩ Span(x2, x3, x4)= (a1x1+a2x2+a3x3) ∩ (a2x2+a3x3+a4x4)=a1x1+a2x2+a3x3+a4x4= Span(x1, x2, x3, x4)≠ Span(x2,x3)
Is this okay?
or is it ok to just do a counter example using vectors,
if so could somebody show me an example counter example
thanks in advance