- #1

DRose87

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**(sorry for the horrible butchered thread title... should say "determination", not "determining")**

1. Homework Statement

1. Homework Statement

In "Principles of Quantum Mechanics", by R. Shankar, a vector space is defined as having dimension n if it can accommodate a maximum of n linearly independent vectors (here is a free and legal preview of the math introduction. The parts I am discussing are on page 5 https://books.google.com/books?id=sDvrBwAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false). Immediately after this definition, the author attempts to prove that the set of all 2x2 matrices forms a four dimensional vector space. I am fully aware that this is the case, but the author's "proof" seems to be leaving out various facts, I think, needed to establish this (unless I am missing something and it should be completely obvious that 2x2 matrices can accommodate a maximum of n linearly independent vectors, without knowing anything of rules concerning spanning sets and bases, with have not been covered by this point in the book). The author states that because the matrices

Code:

```
[tex]\left| 1 \right\rangle = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&0
\end{array}} \right],\left| 2 \right\rangle = \left[ {\begin{array}{*{20}{c}}
0&1\\
0&0
\end{array}} \right],\left| 3 \right\rangle = \left[ {\begin{array}{*{20}{c}}
0&0\\
1&0
\end{array}} \right],\left| 4 \right\rangle = \left[ {\begin{array}{*{20}{c}}
0&0\\
0&1
\end{array}} \right][/tex]
```

I just am wondering if this fact should be totally obvious, since the author just states that 2x2 matrices can accommodate at most four linearly independent vectors because any 2x2 matrix can be written as a combination of the four simple basis vectors shown above... before even mentioning what a basis is or explicitly stating why being able to write any 2x2 matrix as a combination of |1>,|2>,|3>,|4> prohibits the possible existence of five linearly independent vectors.

**2 Attempt At A Solution**

I looked in my notes and found a 'proof' (doubt a mathematician would say this is a valid proof but I'm satisfied with it :) ) that, if applied to a 2x2 vector space would show that it is of dimension 4... but it actually involved working out a lot of details lol... So I am wondering if the author of the book expects students to be familiar enough with linear algebra to know the results of the proof posted below... or if the fact that any set of 5 or more 2x2 matrices will be linearly dependent should be totally obvious considering that the basis posted above (which the author doesn't even call a basis... presumably up to this point in the book he hasn't even mentioned the concept of a basis) consists of four elements.

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