Quick question in linear algebra review for quantum mechanics

[iScience]

so in my book there is an example basically saying that linear transformations can be applied to basis vectors or, more specifically, i think they're using orthonormal basis vectors |e₁>,|e₂>, ...

i'm just a little confused on how they're applying it to the basis vectors.

my book:

http://i.imgur.com/BFNMuz0.jpg

\hat{T}|e₁> ; okay so here on the LHS we're applying the transformation to the first orthonormal basis vector. the first term on the RHS, is this basically saying how much we're extending or contracting along the |e₁> direction?

 

[wotanub]

It could also be a rotation or reflection or a combination of the three and so on. Try thinking about the linear transformation as a map (function) on basis vectors. You input a basis vector and you get out something based on what you put in.

Consider the following transformation

Let's say you have three linearly independent vectors a, b, and c. The transformation T has the following action:

a → a + b
b → c
c → b + c

If T is linear, and we can express any vector as a linear combination of these three vectors, we now the action of T on any vector.

For example
v = 3a + 2b + c
Tv = T(3a + 2b + c) = T(3a) + T(2b) + T(c) = 3T(a) + 2T(b) + T(c) = 3(a+b) + 2(c) + (b+c) = 3a + 4b + 3c

So this is some weird transformation that scales and rotates or something. There are various ways to represent operators and you're probably familiar with representing vectors as column matrices and operators as square matrices (this has many advantages). But in general the most abstract way to define a transformation is with a list of rules. You could figure out the matrix for my transform if you want.

Notice I didn't say anything about orthonormality, but in physics you generally are working in an orthonormal basis.

Of course, the map could be one that maps each basis vector to some combination vectors in a different basis as well.