I am reading Linear Algebra from Hoffman Kunze by myself and I have a question in a theorem of Section-" Invariant Direct Sums " :
I am confused by how authors prove uniqueness of expression of $\alpha$ in proving that direct sum representation exists as for proving that direct sum representation exists I must show that representation of $\alpha$ is unique?
I got confused as I think author should take two representations and then prove them both be equal but here author only took $E_{i} \beta_{i} $ = $\alpha_{i} $ .
Can kindly tell how exactly author proved uniqueness( ie how last 6 lines of image).
Best Answer
They did. They first established that $$\alpha = E_1\alpha + \cdots + E_n\alpha$$ is one desired decomposition with $E_i\alpha \in W_i$.
Then they took a different decomposition $$\alpha = \alpha_1 + \cdots + \alpha_n$$ with $\alpha_i \in W_i$ and showed that actually it must be $\alpha_i = E_i\alpha$ for $1 \le i \le n$.
Therefore the decomposition $\alpha = E_1\alpha + \cdots + E_n\alpha$ is unique. For any other decomposition $$\alpha = \beta_1 + \cdots + \beta_n$$ with $\beta_i \in W_i$ we would also get $\beta_i = E_i\alpha$ and hence $\beta_i = \alpha_i$ for all $1 \le i \le n$.