Page 73 - Linear Algebra Done Right

Exercises
Exercises
1. Show that every linear map from a one-dimensional vector space 59
to itself is multiplication by some scalar. More precisely, prove
that if dim V = 1 and T ∈L(V, V), then there exists a ∈ F such
that Tv = av for all v ∈ V.
2
2. Give an example of a function f : R → R such that Exercise 2 shows that
homogeneity alone is
f(av) = af (v) not enough to imply
that a function is a
2
for all a ∈ R and all v ∈ R but f is not linear.
linear map. Additivity
3. Suppose that V is finite dimensional. Prove that any linear map alone is also not
on a subspace of V can be extended to a linear map on V.In enough to imply that a
function is a linear
other words, show that if U is a subspace of V and S ∈L(U, W),
map, although the
then there exists T ∈L(V, W) such that Tu = Su for all u ∈ U.
proof of this involves
4. Suppose that T is a linear map from V to F. Prove that if u ∈ V advanced tools that are
is not in null T, then beyond the scope of
this book.
V = null T ⊕{au : a ∈ F}.

5. Suppose that T ∈L(V, W) is injective and (v 1 ,...,v n ) is linearly
independent in V. Prove that (Tv 1 ,...,Tv n ) is linearly indepen-
dent in W.
6. Prove that if S 1 ,...,S n are injective linear maps such that S 1 ...S n
makes sense, then S 1 ...S n is injective.

7. Prove that if (v 1 ,...,v n ) spans V and T ∈L(V, W) is surjective,
then (Tv 1 ,...,Tv n ) spans W.

8. Suppose that V is finite dimensional and that T ∈L(V, W). Prove
that there exists a subspace U of V such that U ∩ null T ={0}
and range T ={Tu : u ∈ U}.

4
2
9. Prove that if T is a linear map from F to F such that
4
null T ={(x 1 ,x 2 ,x 3 ,x 4 ) ∈ F : x 1 = 5x 2 and x 3 = 7x 4 },
then T is surjective.