1. ## Linear Transformation

Prove whether or not these functions are linear transformation.

a) $\displaystyle F:V_3(R) \to V_2(R),$ defined by $\displaystyle F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)$

b) $\displaystyle G:V_3(R) \to V_2(R),$ defined by $\displaystyle G(a_1, a_2, a_3)=(a_1-a_2, 2a_1-a_3)$

2. A transformation, f, is linear if and only if it satisfies f(au+ bv)= af(u)+ bf(v) for any vectors u and v and any numbers a and b.

If $\displaystyle F(a_1, a_2, a_3)= (a_1+ a_2, a_1a_3- a_2)$ then what is $\displaystyle F(a_1+ b_1, a_2+ b_2, a_3+ b_3)$. Is it the same as $\displaystyle F(a_1,a_2,a_3)+ F(b_1, b_2, b_3)$?

3. Yeah I don't know how to show whether it is Linear or not.

4. Originally Posted by LL_5
Yeah I don't know how to show whether it is Linear or not.
You have been told what to do. Try answering the question you were asked in post #2. Where do you get stuck?

5. This is what I have done so far to prove whther it is linear:

6. $\displaystyle F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)$

Let $\displaystyle v_1= (x_1,y_1,z_1)$ and $\displaystyle v_2=(x_2,y_2,z_2)$

A map is linear if $\displaystyle F(\alpha_1 v_1+\alpha_2 v_2)=\alpha_1F(v_1)+\alpha_2F(v_2)$

$\displaystyle F \left( \alpha_1 (x_1,y_1,z_1)+\alpha_2(x_2,y_2,z_2) \right)=F(\alpha_1 x_1+ \alpha_2 x_2, \alpha_1y_1+\alpha_2y_2, \alpha_1 z_1+ \alpha_2 z_2)$$\displaystyle =(\alpha_1 x_1+\alpha_2 x_2 +\alpha_1 y_1+\alpha_2 y_2, \alpha_1^2 x_1 z_1 + \alpha_2 \alpha_1 x_2 z_1+ \alpha_1 \alpha_2 x_1 z_2 + \alpha_2^2 x_2 z_2-\alpha_1y_1-\alpha_2y_2)$

$\displaystyle \alpha_1F(v_1)+\alpha_2F(v_2)=\alpha_1(x_1+y_1, x_1 z_1-y_1)+ \alpha_2 (x_2+y_2, x_2 z_2-y_2)$

But $\displaystyle F(\alpha_1 v_1+\alpha_2 v_2) \neq \alpha_1F(v_1)+\alpha_2F(v_2)$ so $\displaystyle F$ is not a linear transformation.

Now you try and do $\displaystyle G$. I think it is linear.....