Talk:Linear Algebra/Dimension/Solutions

Problem 7 is ambiguous
It asks for dimension of $$\mathbb{C}^{47}$$ but that depends on whether the space is over $$\mathbb{R}$$ in which case the answer is 2*47, or if the space is over $$\mathbb{C}$$ in which case the answer is 47. Orangus (discuss • contribs) 23:46, 7 June 2021 (UTC)

Mistake/mislead in answer to Problem 21
It seems that in the explanation all $$\vec{e}_3 - \vec{e}_2$$ should have been $$\vec{e}_3 - \vec{e}_1$$. That makes it clear why $$\vec{e}_2 - \vec{e}_1,\, \vec{e}_3 - \vec{e}_1,\,\dots,\,\vec{e}_n - \vec{e}_1$$ is linearly independent. It is unclear why $$\vec{e}_2 - \vec{e}_1,\, \vec{e}_3 - \vec{e}_2,\,\dots,\,\vec{e}_n - \vec{e}_1$$ is linearly indendent at all, since that looks to mean $$\vec{e}_2 - \vec{e}_1,\, \vec{e}_3 - \vec{e}_2,\, \vec{e}_4 - \vec{e}_3,\, \vec{e}_5 - \vec{e}_4,\, \dots,\, \vec{e}_n - \vec{e}_{n-1},\, -(\vec{e}_1 - \vec{e}_n)$$ which is not linearly independent, just add up all terms but the last, they cancel out nicely, and you get a multiple of the last term.

Also explains where all the $$x$$'s between $$x_2$$ and $$x_n$$ are coming from in $$\vec{v} = x_1\vec{e}_1 + x_2\vec{e}_2 + \cdots + x_n\vec{e}_n = (x_1 + x_2 + \underset{\text{here}}{\cdots} + x_n)\vec{e}_1 + x_2(\vec{e}_2 - \vec{e}_1) + \cdots + x_n(\vec{e}_n - \vec{e}_1) $$.

I am not going to edit it but just note that here, because this is a Problem that is marked with " ? ", which in the beginning of the book is explained that problems marked with " ? " are cited(both the statement and the solution) as close to verbatim as possible. Orangus (discuss • contribs) 22:46, 7 June 2021 (UTC)