How Do You Prove the Variance Formula for a Linear Combination of Variables?

[cse63146]

Homework Statement

Let $$\Sigma =$$
( var(X1) cov(X1, X2) )
( cov (X2. X1) var(X2) )

Show that $$Var (a_1 X_1 + a_2 X_2) = a^T \Sigma a$$

where $$a^T = [a_1 a_2]$$ is the transpose of the of the column vector a

Homework Equations

The Attempt at a Solution

I got this far:

$$Var (a_1 X_1 + a_2 X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + 2a_1 a_2 Cov (X_1, X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + a_1 a_2 Cov (X_1, X_2) + a_1 a_2 Cov (X_2, X_1)$$

Thats all I got so far, any hints

 

[Billy Bob]

Aren't you done? Isn't that what $$a^T \Sigma a$$ is?

 

[cse63146]

Thought there was more to it than that.

There's another part of the question that says: Using $$Var (a_1 X_1 + a_2 X_2)$$ show that for every choice of a1 and a2 that $$a^T \Sigma a \geq 0$$

Can I assume that $$\Sigma$$ is always positive?

 

[Billy Bob]

$$Var (a_1 X_1 + a_2 X_2)\ge0$$ always, since it's variance! And you just showed it equals $$a^T \Sigma a$$