Page 299 - 35Linear Algebra

16.4 Review Problems 299

16.4 Review Problems

Reading Problems 1 , 2 ,
Elements of kernel 3
Basis for column space 4
Basis for kernel 5
Basis for kernel and range 6
Webwork:
Orthonomal range basis 7
Orthonomal kernel basis 8
Orthonomal kernel and range bases 9
Orthonomal kernel, range and row space bases 10
Rank 11

m
n
1. Consider an arbitrary matrix M : R → R .
(a) Argue that Mx = 0 if only if x is perpendicular to all columns
T
of M .
(b) Argue that Mx = 0 if only if x is perpendicular to all of the linear
T
combinations of the columns of M .
T
(c) Argue that ker M is perpendicular to ran M .
m
T
(d) Argue further R = ker M ⊕ ran M .
T
n
(e) Argue analogously that R = ker M ⊕ ran M.
The equations in the last two parts describe how a linear transforma-
n
m
tion M : R → R determines orthogonal decompositions of both it’s
domain and target. This result sometimes goes by the humble name
The Fundamental Theorem of Linear Algebra.

2. Let L: V → W be a linear transformation. Show that ker L = {0 V } if
and only if L is one-to-one:

(a) (Trivial kernel ⇒ injective.) Suppose that ker L = {0 V }. Show
that L is one-to-one. Think about methods of proof–does a proof
by contradiction, a proof by induction, or a direct proof seem most
appropriate?
(b) (Injective ⇒ trivial kernel.) Now suppose that L is one-to-one.
Show that ker L = {0 V }. That is, show that 0 V is in ker L, and
then show that there are no other vectors in ker L.

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