Exercise 18

The material in this subsection allows us to express a geometric relationship that we have not yet seen between the range space and the null space of a linear map.

• Represent $f\colon\mathbb{R}^{3}\to\mathbb{R}$ given by

  $$\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\end{pmatrix}\mapsto 1v_{1}+2v_{2}+3v_{3}$$

  with respect to the standard bases and show that

  $$\begin{pmatrix}1\\ 2\\ 3\end{pmatrix}$$

  is a member of the perp of the null space. Prove that $\mathscr{N}(f)^{\perp}$ is equal to the span of this vector.

• Generalize that to apply to any $f\colon\mathbb{R}^{n}\to\mathbb{R}$.

• Represent $f\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$

  $$\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\end{pmatrix}\mapsto\begin{pmatrix}1v_{1}+2v_{2}+3v_{3}\\ 4v_{1}+5v_{2}+6v_{3}\end{pmatrix}$$

  with respect to the standard bases and show that

  $$\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\;\begin{pmatrix}4\\ 5\\ 6\end{pmatrix}$$

  are both members of the perp of the null space. Prove that $\mathscr{N}(f)^{\perp}$ is the span of these two. (Hint. See the third item of Exercise 17.)

• Generalize that to apply to any $f\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$.

In this is called the Fundamental Theorem of Linear Algebra