Will random sampling from image of a matrix span the range of it?

Question

Suppose I have a matrix $A$, $A$ has full rank of 3. Now lets assume I have three linearly independent vectors namely $x_{1}$, $x_2$, $x_3$, so I have $$y_1=Ax_1$$ $$y_2=Ax_2$$ $$y_3=Ax_3$$ Now if $y_1$,$y_2$ and $y_3$ are also linearly independent, will they span image of A?

Answer 1

Since $x_1,x_2,x_3$ are lineraly independent then $$a_1x_1+a_2x_2+a_3x_3=0\iff a_1=a_2=a_3=0.$$ Thus

$$a_1y_1+a_2y_2+a_3y_3=A(a_1x_1+a_2x_2+a_3x_3)=0 \iff a_1=a_2=a_3=0,$$ which shows that $y_1,y_2,y_3$ are linearly independent.

Since they are lineraly independent they form a basis of $\mathbb{R}^3=\rm{im}(A)$. Thus, every vector of $\mathbb{R}^3$ is a linear combination of $y_1,y_2,y_3.$