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Linear Algebra

Exercise 3.2.14

Answers

1.

For all

y \in R (T + U)

we can express it as

y = T (x) + U (x) \in R (T) + R (U)

for some

x \in V

.

2.

By Theorem 1.6.29(a) we have

rank (T + U) \leq \dim (R (T) + R (U))

= \dim (R (T)) + \dim (R (U)) - \dim (R (T) \cap R (U))

\leq rank (T) + rank .

3.

We have that

rank (A + B) = rank (L_{A + B}) = rank (L_{A} + L_{B})

\leq rank (A) + rank (B) .

jephianlin

2011-06-27 00:00

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