I want to check of the following functions are linear or not.
$g:\mathbb{R}^2\to \mathbb{R}, \ (x,y)\mapsto x+3y$
$h:\mathbb{R}^3\to \mathbb{R}^2, \ (x,y,z)\mapsto (x+y+z, 2y-z)$
I have done the following :
Let $(x_1,y_1), (x_2,y_2)\in \mathbb{R}^2$. Then we have \begin{align*}g\left ((x_1,y_1)+(x_2,y_2)\right )&=g(x_1+x_2,y_1+y_2)\\ & =(x_1+x_2)+3(y_1+y_2)=(x_1+3y_1)+(x_2+3y_2) \\ & =g(x_1,y_1)+g(x_2,y_2)\end{align*} Let $\lambda\in \mathbb{R}$. Then we have \begin{align*}g\left (\lambda (x,y)\right )&=g(\lambda x, \lambda y)\\ & =\lambda x+3\lambda y\\ & =\lambda (x+3y)\\ & =\lambda g(x,y)\end{align*} Therefore $g$ is linear.
Let $(x_1,y_1,z_1), (x_2,y_2,z_2)\in \mathbb{R}^3$. Then we have \begin{align*}h\left (c+(x_2,y_2,z_2)\right )&=h(x_1+x_2,y_1+y_2,z_1+z_2)\\ &=\left ((x_1+x_2)+(y_1+y_2)+(z_1+z_2), 2(y_1+y_2)-(z_1+z_2)\right )\\ & =\left ((x_1+y_1+z_1)+(x_2+y_2+z_2), (2y_1-z_1)+(2y_2-z_2)\right )\\ & =(x_1+y_1+z_1,2y_1-z_1)+(x_2+y_2+z_2,2y_2-z_2)\\ & =h(x_1,y_1,z_1)+h(x_2,y_2,z_2)\end{align*} Let $\lambda\in \mathbb{R}$. Then we have \begin{align*}h\left (\lambda (x,y,z)\right ) & =h(\lambda x, \lambda y,\lambda h)\\ & =(\lambda x+\lambda y+\lambda z, 2\lambda y-\lambda z)\\ & =\lambda (x+y+z,2y-z)\\ & =\lambda h(x,y,z)\end{align*} Therefore $h$ is linear.
Is everything correct? Or am I missing something?
This is correct, except for when you started 2. with $h(c+(x2,y2,z2))$ instead of $h((x1,y1,z1)+(x2,y2,z2))$.