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I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

I need help with some aspects of the proof of Lemma 1.2 ... ...

Lemma 1.2 reads as follows:

My questions related to the above proof by Bresar are as follows:

In the above text from Bresar we read the following:

" ... ... Since ##u, v, 1## are linearly independent, this yields ##\lambda + \mu = \lambda - \mu = 0##, hence ##\lambda = \mu = 0##, and ##u + v \in V## follows from the first paragraph. ... ... "My question is ... ... how

In the above text from Bresar we read the following:

" ... ... Again, using the observation from the first paragraph we see that ##x + \frac{v}{2} \in V##. Accordingly, ##x = - \frac{v}{2} + ( x + \frac{v}{2} ) \in \mathbb{R} \oplus V##. ... ... "My question is ... ... how

Hope someone can help ...

Peter

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In order for readers of the above post to appreciate the context of the post I am providing pages 1-2 of Bresar ... as follows ...

I need help with some aspects of the proof of Lemma 1.2 ... ...

Lemma 1.2 reads as follows:

My questions related to the above proof by Bresar are as follows:

**Question 1**In the above text from Bresar we read the following:

" ... ... Since ##u, v, 1## are linearly independent, this yields ##\lambda + \mu = \lambda - \mu = 0##, hence ##\lambda = \mu = 0##, and ##u + v \in V## follows from the first paragraph. ... ... "My question is ... ... how

*does it follow that ##u + v \in V##?***exactly****Question 2**In the above text from Bresar we read the following:

" ... ... Again, using the observation from the first paragraph we see that ##x + \frac{v}{2} \in V##. Accordingly, ##x = - \frac{v}{2} + ( x + \frac{v}{2} ) \in \mathbb{R} \oplus V##. ... ... "My question is ... ... how

*does it follow that ##x + \frac{v}{2} \in V## and, further, how***exactly***does it then follow that ##x = - \frac{v}{2} + ( x + \frac{v}{2} ) \in \mathbb{R} \oplus V## ... ... ?***exactly**Hope someone can help ...

Peter

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In order for readers of the above post to appreciate the context of the post I am providing pages 1-2 of Bresar ... as follows ...