Question

2. Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$. (a) Prove that null$T$ is invariant under $T$. (b) Prove that range$T$ is invariant under $T$. (c) Prove that if $U \subset$ null$T$, then $U$ is invariant under $T$. (d) Prove that if range$T \subset U$, then $U$ is invariant under $T$.

          2. Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V$.
(a) Prove that null$T$ is invariant under $T$.
(b) Prove that range$T$ is invariant under $T$.
(c) Prove that if $U \subset$ null$T$, then $U$ is invariant under $T$.
(d) Prove that if range$T \subset U$, then $U$ is invariant under $T$.

2. Suppose T ∈ℒ(V) and U is a subspace of V.
(a) Prove that nullT is invariant under T.
(b) Prove that rangeT is invariant under T.
(c) Prove that if U ⊂ nullT, then U is invariant under T.
(d) Prove that if rangeT ⊂ U, then U is invariant under T.

Added by Ray H.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Ramesh Rajak

Step 1

Let x be a vector in null T. This means that T(x) = 0, since x is mapped to the zero vector by T. Now, let's apply T to T(x): T(T(x)) = T(0) = 0 Since T(T(x)) = 0, this means that T(x) is also in null T. Show more…

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