I am self-studying linear algebra from the book Linear Algebra done right 2nd ed. These are 3 statements about UTM:
Suppose (an operator) T $\in$ L(V, V) and $(v_1, v_2, \ldots, v_n)$ is a basis of V. The following statements are equivalent:
- the matrix M(T) with respect to the basis $(v_1, \ldots, v_n)$ is upper triangular;
- $Tv_k \in span(v_1, \ldots, v_k)$ for each $k=1, 2, \ldots, n$;
- $span(v_1, \ldots, v_k)$ is invariant under T for each $k=1, 2, \ldots, n$;
The book says "The equivalence of 1 and 2 follows easily from the definition since 2 implies that the matrix elements below the diagonal are zero". I really can't figure out why 2 implies 1 or vice versa. Suppose $Tv_k \in span(v_1, \ldots, v_k)$. Then $Tv_1 = a_1v_1$, $Tv_2 = b_1v_1 + b_2v_2$, $\ldots$. When applied the operator to polynomial, Fundamental Theorem of Algebra says every operator has at least an eigenvalue, so $Tv_1 = a_1v_1$ gives no info about the matrix. How do we get to the conclusion that T must be upper triangular from this point? Thanks.