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.
(a) Represent f : R3 †’ R given by

with respect to the standard bases and show that

is a member of the perp of the null space. Prove that N (f) ”´ is equal to the span of this vector.
(b) Generalize that to apply to any f : Rn †’ R.
(c) Represent f : R3 †’ R2

with respect to the standard bases0and show that

are both members of the perp of the null space. Prove that N (f)”´ is the span of these two.
(d) Generalize that to apply to any f : Rn †’ Rm.
In [Strang 93] this is called the Fundamental Theorem of Linear Algebra

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