How do change of basis matrices work in linear algebra?

[aaaa202]

Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors. I'm finding this a little hard to understand intuitively - can someone give me an example from maybe R^2 that shows why this is in an intuitive manner.

 

[Stephen Tashi]

that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors.

That depends on how we define a "change of basis matrix". For example if I have the row vector $V_a$ in basis $A$ and $S$ is "the change of basis matrix from basis $A$ to basis $B$ then how do I find that vector expessed in basis $B$?

Do I compute $(V_a)( S)$ or do I compute $(S)(V_a)^T$ ? It depends on whether the course materials like to treat vectors as row vectors or as column vectors.

You should be able to construct a 2x2 example.

Let
$a_1 = S_{11} b_1 + S_{12} b_2$
$a_2 = S_{21} b_1 + S_{22} b_2$

What is the vector $4 a_1 + 3 a_2$ equal to in the $B$ basis?

 

[chiro]

You can factor out transformations in terms of rotations, translations, and scaling operations.

By composing maps, you can see geometrically what the translation is doing by treating each map separately in terms of its geometric effect on a vector.

 

[lavinia]

Given a basis A = {a1,a2...an} we can always translate coordinates originally expressed with this basis to another basis A' = {a1',a2'...an'}. To do this we simply do some matrix-multiplication and it turns out that the change of basis matrix equals a square matrix whose rows are the coordinates of the original basis vectors written in terms of the new basis-vectors. I'm finding this a little hard to understand intuitively - can someone give me an example from maybe R^2 that shows why this is in an intuitive manner.

Each vector in the first basis can be written as a linear combination of the vectors in the second. The coefficients of this linear combination are a column in the change of basis matrix.
The set of columns, one for each basis vector in the first basis, make up the change of basis matrix.
Multiplying a basis vector by the matrix just writes the vector out in terms of the second basis.

E.g. take for the first basis of R^2 the vectors (1,0) and (0,1) and the second basis (1,-1) and (1.1).

Then (1,0) = 1.2 (1,-1) + 1/2(1,1) so the first column of the matrix is (1/2 1/2).

The second column is (-1/2 1/2)