Let $\mathbb{R}$ denote the set of real numbers.
Consider the function $f$ on $R \times R$ defined by $f(x,y) = (x+y, 2x-3y)$.
Is $f$ onto? justify your answer.
Let $\mathbb{R}$ denote the set of real numbers.
Consider the function $f$ on $R \times R$ defined by $f(x,y) = (x+y, 2x-3y)$.
Is $f$ onto? justify your answer.
HINT $\ $ Since $\rm\:f\:$ is $\rm\:\mathbb R$-linear, i.e. $\rm\ f(a\ u + b\ v)\ =\ a\ f(u) + b\ f(v)\ $ for $\rm\ a,b\in \mathbb R,\ \ u,v\in \mathbb R^2\:,\:$ its image $\rm\:f(\mathbb R^2)\:$ spans $\rm\:\mathbb R^2\:$ if the image includes a basis, say $\rm\ (0,1),\ (1,0)\:.\:$ This is easy to verify, viz.
$\rm\qquad\qquad\quad\ v = (\:r,\:-r)\ \Rightarrow\ f(v)= (0,5r)\ $ so $\rm\ r = 1/5\ \Rightarrow\ f(v) = (0,1) $
$\rm\qquad\qquad\quad u = (3r,2r)\ \Rightarrow\ f(u)= (5r,0)\ $ so $\rm\ r = 1/5\ \Rightarrow\ f(u) = (1,0) $
Hence $\rm\ (a,b) = a\ (1,0) + b\ (0,1)\ =\ a\ f(u) + b\ f(v)\ =\ f(au)+f(bv)\ =\ f(au+bv)\:.$
For an analogous example look up the standard proof of the Chinese Remainder Theorem (CRT). It too uses linearity to construct the solution for $\rm\:(a,b)\:$ from the solutions for $\rm\:(1,0)\:$ and $\rm\:(0,1)\:.$
Hint: does the system of equations $ \eqalign{x+y&=a\cr 2x-3y&=b} $have a solution for every $(a,b)$?
If $f(x,y) = (x+y, 2x-3y)$ has an inverse $\left(\dfrac{3a+b}{5},\dfrac{2a-b}{5}\right)=g(a,b)$ then not only is $g$ the inverse of $f$ but $f$ is the inverse of $g$. So $f$ is bijective: both one-to-one and onto.
We know $f$ is onto because we can obtain any $a$ and $b$ using $f\left(\dfrac{3a+b}{5},\dfrac{2a-b}{5}\right)=(a,b). $