Determine if Linear x([[a],[b],[d]])=[[2a+d],[-4b]]

Problem

L*([a],[b],[d])=[[2*a+d],[−4*b]]

Solution

1. Identify the definition of a linear transformation. A transformation L:V→W is linear if it satisfies two properties for all vectors u,v and scalars c L*(u+v)=L(u)+L(v) and L*(c*u)=c*L(u)

2. Test additivity by defining two vectors u=[[(a_1)],[(b_1)],[(d_1)]] and v=[[(a_2)],[(b_2)],[(d_2)]]

3. Compute L*(u+v)

L*(u+v)=L*([(a_1)+(a_2)],[(b_1)+(b_2)],[(d_1)+(d_2)])

L*(u+v)=[[2*((a_1)+(a_2))+((d_1)+(d_2))],[−4*((b_1)+(b_2))]]

L*(u+v)=[[2*(a_1)+2*(a_2)+(d_1)+(d_2)],[−4*(b_1)−4*(b_2)]]

4. Compute L(u)+L(v)

L(u)+L(v)=[[2*(a_1)+(d_1)],[−4*(b_1)]]+[[2*(a_2)+(d_2)],[−4*(b_2)]]

L(u)+L(v)=[[2*(a_1)+(d_1)+2*(a_2)+(d_2)],[−4*(b_1)−4*(b_2)]]

⇒L*(u+v)=L(u)+L(v)

5. Test homogeneity by multiplying a vector by a scalar c

L*(c*u)=L*([c*(a_1)],[c*(b_1)],[c*(d_1)])

L*(c*u)=[[2*(c*(a_1))+(c*(d_1))],[−4*(c*(b_1))]]

L*(c*u)=[[c*(2*(a_1)+(d_1))],[c*(−4*(b_1))]]

L*(c*u)=c*[[2*(a_1)+(d_1)],[−4*(b_1)]]

⇒L*(c*u)=c*L(u)

6. Conclude that since both properties of linearity are satisfied, the transformation is linear.

Final Answer

L*([a],[b],[d])=[[2*a+d],[−4*b]]* is linear

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