Thread: Linear combination is also in the span

Suppose that vectors x and y are both in the linear span of the
vectors r1, r2, r3. Show that any linear combination of x and y is also
in the linear span of r1, r2, r3.

What I have-

Because x and y are in the span of r1,r2,r3, then x and y are linear combinations of r1,r2,r3. So, the following exists:

c1(r1)+c2(r2)+c3(r3) = x

and

d1(r1)+d2(r2)+d3(r3) = y

where the d's and c's are scalars. I'm not sure this gets me anywhere with the proof.

Thanks.

if:

, and ,

then



which is surely in span({r₁,r₂,r₃})

since for each i, is surely some scalar.

Originally Posted by sfspitfire23

Suppose that vectors x and y are both in the linear span of the vectors r1, r2, r3. Show that any linear combination of x and y is also
in the linear span of r1, r2, r3.

Suppose that .

Show that .

i.e.