2 questions in text of Lesson : Inner Product spaces of Hoffman Kunze Linear Algebra( Related to Orthogonality)

Question

I am self studying linear Algebra from Hoffman Kunze and I have 2 questions in text given just after Corollary of Theorem 3 whose image I am adding below.

It's image:

Question (1): Why in last line of page 281 we must have $\alpha_{m+1} $ =0?

Question 2: How does in 2nd last line, $\alpha_{1}$ , ... $\alpha_{m} $ being different from 0 and $\alpha_{m+1} $ = 0 implies $\beta_{i}'s$ to be linearly dependent?

For (8-9 ) see image :

I shall be really thankful for any explanations provided.

Answer 1

For Question 1: since $\alpha_{m+1}$ is in the span of the orthogonal basis $\alpha_1,\ldots,\alpha_m$, you can express $\alpha_{m+1}$ using (8-8): $$\alpha_{m+1}=\sum_{k=1}^m\frac{(\alpha_{m+1}|\alpha_k)}{\|\alpha_k\|^2}\alpha_k$$ But $(\alpha_{m+1}|\alpha_k)=0$ for all $k$, so $\alpha_{m+1}=0$.

For Question 2: under these assumptions, equation (8-9) gives you a nontrivial linear dependence relation between the $\beta_i$'s, so they are linearly dependent.