| # Fundamental theorem of linear algebra Fundamental theorem of linear algebra Let . The sets and form an orthogonal decomposition of , in the sense that any vector can be written as In particular, we obtain that the condition on a vector to be orthogonal to any vector in the nullspace of implies that it must be in the range of its transpose: Proof: The theorem relies on the fact that if a SVD of a matrix is then an SVD of its transpose is simply obtained by transposing the three-term matrix product involved: Thus, the left singular vectors of are the right singular vectors of . From this we conclude in particular that the range of is spanned by the first columns of . Since the nullspace of is spanned by the last columns of , we observe that the nullspace of and the range of are two orthogonal subspaces, whose dimension sum to that of the whole space. Precisely, we can express any given vector in terms of a linear combination of the columns of ; the first columns correspond to the vector and the last to the vector : This proves the first result in the theorem. The last statement is then an obvious consequence of this first result: if is orthogonal to the nullspace, then the vector in the theorem above must be zero, so that .