Image of a linear map ($\mathrm{im}T$)

The image of a linear map $T:V\to W$ is the vector subspace $\mathrm{im}T=\{T(v):v\in V\}$ of its codomain.

For a real matrix $A$ and a fixed vector $n$, every vector in the image of a linear map defined by $A$ is orthogonal to $n$ exactly when $A^{T}n=0$. Indeed, $n^{T}Ar=(A^{T}n{)}^{T}r$. If this vanishes for every $r$, choose $r=A^{T}n$ to obtain $|A^{T}n{|}^2=0$. Equivalently each column of $A$ is orthogonal to $n$. Requiring the statement only for nonzero $r$ changes nothing, since the zero input already has zero dot product. This is the orthogonal complement relation between the image of a linear map and the kernel of a linear map defined by its transpose.

If a linear functional $\ell$ vanishes on the image of a linear map $T$ but $\ell (b)\ne 0$, the equation $Tx=b$ has no solution. In Euclidean coordinates, a vector $n$ with $n^{T}A=0$ provides such an obstruction whenever $n^{T}b\ne 0$. In finite dimensions these annihilating functionals exactly characterize membership in the image: $b\in \mathrm{im}A$ precisely when it is orthogonal to every vector in $\ker A^T$.