Proof of the conditions holds: if c is any real number and u is any vector in w, than c .u is in w?

Question

c.u, is c dot u, or c times u. in subspaces vector

Answer 1

Suppose the subspace w = span{w1,w2,...wn} where the vectors w1, ... , wn form a basis for w.

Since u is in w, u is a linear combination of these vectors.

u = a1 w1 + a2 w2 + ... + an wn
where a1,...,an are the weights.

multiply this equation by c to get:
cu = (c a1) w1 + (ca2) w2 + ... + (c an) wn

This shows that cu is a linear combination of the vectors in w = span{w1,w2,...wn} using weights (c a1),...,(c an).
By definition cu is in w.