Given 2 planes such that $R(A)=R([A,b])=2$,how to show that the solution is a line?

Question

Suppose we are given two linear equations in $x_1,x_2,x_3$ given by $Ax=b$ where $A$ is $2\times3$ matrix and $b$ is $2\times1$ matrix such that $rank(A)=rank(A,b)=2$,then clearly the system has infinitely many solutions but how to show that the solution set is a line?I have done it by one method,suppose the system is $a_1x_1+a_2x_2+a_3x_3=b$,without loss let $a_1,a_2$ form a basis of the column space of $[A,b]$.then express $a_3$ and $b$ in terms of the two basis vectors.Take all in left side with $0$ at right side in the given eqns after replacing the other two vectors as the L.C. of the basis vectors.Since $a_1$ and $a_2$ are L.I.,hence we get a relation between the 3 variables,which is the equation of a line.But is there any other way to prove that it will be a line?

Answer 1

Consider the associated linear system $Ax=0$. By hypothesis, $\mathrm{rank}(A) =2$. $A$ is a linear operator from $\mathbb R^3$ onto $\mathbb R^2$. By the rank-nullity theorem, $\dim \ker A = \dim \mathbb R^3 - \dim \mathrm{rank}(A) =1$.

The solutions of $Ax=b$ lie in an affine space whose associated vector space ($\ker A$) is a vectorial line. Therefore this affine space is an affine line.