3A.17

Suppose $𝑉$ is finite-dimensional. Show that the only two-sided ideals of $ℒ(𝑉)$ are $\{0\}$ and $ℒ(𝑉)$.

A subspace $ℰ$ of $ℒ(𝑉)$ is called a two-sided ideal of $ℒ(𝑉)$ if $𝑇𝐸 ∈ ℰ$ and $𝐸𝑇 ∈ ℰ$ for all $𝐸 ∈ ℰ$ and all $𝑇 ∈ ℒ(𝑉)$.

3B.25

Suppose that 𝑊 is finite-dimensional and 𝑆, 𝑇 ∈ ℒ(𝑉, 𝑊). Prove that null 𝑆 ⊆ null 𝑇 if and only if there exists 𝐸 ∈ ℒ(𝑊) such that 𝑇 = 𝐸𝑆.

3F.25

(b) Prove that $\varphi_1, …, \varphi_𝑚$ is linearly independent if and only if $Γ$ is surjective.

$\Rightarrow$

5A.39

Suppose $𝑉$ is finite-dimensional and $𝑇 ∈ ℒ(𝑉)$. Prove that $𝑇$ has an eigenvalue if and only if there exists a subspace of $𝑉$ of dimension $\dim 𝑉 − 1$ that is invariant under $𝑇$.

$\Rightarrow$.

5A.43

The second strategy is not working yet.

5B.9

Suppose $𝑇 ∈ ℒ(𝑉)$ is such that with respect to some basis of $𝑉$, all entries of the matrix of $𝑇$ are rational numbers. Explain why all coefficients of the minimal polynomial of $𝑇$ are rational numbers.

I can think the $T$ as a linear operator on some vector space over $\mathbb{Q}$, then the minimal polynomial of $T$ has rational coefficients.

Also $p(T) = 0$, if we consider it apply on the standard basis. So $p$ is a multiple of the minimal polynomial.

$\square$