Page 74 - Linear Algebra Done Right

Chapter 3. Linear Maps
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2
5
10.
Prove that there does not exist a linear map from F to F whose
null space equals
5
{(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) ∈ F : x 1 = 3x 2 and x 3 = x 4 = x 5 }.
11. Prove that if there exists a linear map on V whose null space and
range are both finite dimensional, then V is finite dimensional.
12. Suppose that V and W are both finite dimensional. Prove that
there exists a surjective linear map from V onto W if and only if
dim W ≤ dim V.

13. Suppose that V and W are finite dimensional and that U is a
subspace of V. Prove that there exists T ∈L(V, W) such that
null T = U if and only if dim U ≥ dim V − dim W.

14. Suppose that W is finite dimensional and T ∈L(V, W). Prove
that T is injective if and only if there exists S ∈L(W, V) such
that ST is the identity map on V.

15. Suppose that V is finite dimensional and T ∈L(V, W). Prove
that T is surjective if and only if there exists S ∈L(W, V) such
that TS is the identity map on W.

16. Suppose that U and V are finite-dimensional vector spaces and
that S ∈L(V, W), T ∈L(U, V). Prove that

dim null ST ≤ dim null S + dim null T.

17. Prove that the distributive property holds for matrix addition
and matrix multiplication. In other words, suppose A, B, and C
are matrices whose sizes are such that A(B + C) makes sense.
Prove that AB + AC makes sense and that A(B + C) = AB + AC.

18. Prove that matrix multiplication is associative. In other words,
suppose A, B, and C are matrices whose sizes are such that
(AB)C makes sense. Prove that A(BC) makes sense and that
(AB)C = A(BC).