Suppose v⃗ ,w⃗ are linear combinations of u⃗ 1 and u⃗ 2. Prove that 2v⃗ +3w⃗ is also a linear combination of u⃗ 1 and u⃗ 2.

Question

Question

I was just browsing through some linear algebra questions and stumbled on this question. I have no idea how to prove this. I understand what a linear combination is, but how do I prove this properly?

Answer 1

$$v = c_1u_1+c_2u_2,\qquad w = d_1u_1+d_2u_2$$

$$2v+3w = 2(c_1u_1+c_2u_2)+3(d_1u_1+d_2u_2)$$

$$= 2c_1u_1+2c_2u_2+3d_1u_1+3d_2u_2$$

$$= (2c_1+3d_1)u_1+(2c_2+3d_2)u_2$$