Question in proof of a lemma related to Characterstic value of Operator on subspaces I have a question about the proof of a lemma in Hoffman and Kunze's linear algebra text. Here is its image.
Question. How does the highlighted line prove what is asked, i.e., $\dim W = \dim W_{1}+ \cdots + \dim W_{k}?$
Edit: I am adding a related result in which I have a question.
Question. How does the degree of the characteristic polynomial $f$ equal $\dim V?$
Also, in the second-to-last line of this theorem, the authors write $\dim W_{1} + \cdots + \dim W_{k} = \dim V$ implies that $V = W_{1} + \cdots + W_{k}$ by the previous lemma (see the above image), but the statement of the previous lemma is the inverse of the deduction used by authors. So, how did authors derive it?
Please help me understand these questions.
 A: On your first question, if $\beta_1 + \cdots + \beta_k = 0$ implies that $\beta_i = 0$ for each integer $1 \leq i \leq k,$ then the zero vector has a unique representation in $W,$ from which it follows (Exercise 1a) that $W = W_1 \oplus \cdots \oplus W_k.$ Consequently, we have that $\dim W = \dim W_1 + \cdots + \dim W_k,$ as desired.
On the question about the characteristic polynomial of $V,$ we will assume that $\dim V = n.$ Observe that for any linear operator $T$ on $V$ and any ordered basis $\mathscr B$ of $V,$ there exists an $n \times n$ matrix $A$ such that $T(v) = Av$ for every vector $v$ of $V$ written with respect to the basis $\mathscr B.$ We refer to $A$ as the matrix of representation of $T$ with respect to $\mathscr B.$ (For more information about this relationship, see Section 3.4 of Hoffman and Kunze.) Consequently, the characteristic polynomial of $T$ is the same as the characteristic polynomial of $A,$ i.e., $f_T(x) = \det(xI - A).$ Observe that the degree of this polynomial is equal to $\dim V = n$ because $A$ is an $n \times n$ matrix.
Last but not least, if $\dim W_1 + \cdots + \dim W_k = \dim V$ and $W_1 \oplus \cdots \oplus W_k \subseteq V,$ then $V = W_1 \oplus \cdots \oplus W_k.$ One can prove in general that if $W \subseteq V$ and $\dim W = \dim V,$ then $W = V.$
