Linear Algebra, 2020a

294 Chapter Three. Maps Between Spaces
are both members of the perp of the null space. Prove that N (f) ⊥ is the span
of these two. (Hint. See the third item of Exercise 26.)
(d) Generalize that to apply to any f: R n → R m .
In [Strang 93] this is called the Fundamental Theorem of Linear Algebra
3.28 Define a projection to be a linear transformation t: V → V with the property
that repeating the projection does nothing more than does the projection alone: (t◦
t)(⃗v) =t(⃗v) for all ⃗v ∈ V.
(a) Show that orthogonal projection into a line has that property.
(b) Show that projection along a subspace has that property.
(c) Show that for any such t there is a basis B = 〈⃗β 1 ,...,⃗β n 〉 for V such that
{
β ⃗ i i = 1,2,..., r
t(⃗β i )=
⃗0 i = r + 1, r + 2,..., n
where r is the rank of t.
(d) Conclude that every projection is a projection along a subspace.
(e) Also conclude that every projection has a representation
( )
I Z
Rep B,B (t) =
Z Z
in block partial-identity form.
3.29 A square matrix is symmetric if each i, j entry equals the j, i entry (i.e., if the
matrix equals its transpose). Show that the projection matrix A(A T A) −1 A T is
symmetric. [Strang 80] Hint. Find properties of transposes by looking in the index
under ‘transpose’.