True or False: Say $A$ is a $3 \times 7$ matrix such that $Ax=b$ is solvable for all $B$ in $\mathbb{R}^3$. Then $A$ has rank $3$.
I know that rank is the number of non zero rows when the matrix is in RREF which is the same as the number of pivots but I'm not sure how to apply this to an example like this one that has no given values.
On
Suppose rank of $A$ is less than $3$, then there exists a vector in $\mathbb{R}^3$ that is not in the column space of $A$, hence there is no solution. Also rank of $A$ is at least $3$, hence its rank must be exactly $3$.
Now, suppose you are not familiar with concept like column space. Consider RREF of $[A|b]$, let say it is $[R|r]$. Suppose there is always a solution, it means $R$ does not have a zero row, and since each row has a pivot column, the rank is $3$.
The rank is the dimension of the image of the linear map $x\mapsto Ax$. If $Ax=b$ is always solvable, this means that the image if the whole space.