Basics of multiplication of matrices

[Castilla]

Hello, guys. Last week I have begin my study of linear algebra from cero. I am learining about the basics of matrices.

Please, I would like to know if this is ok:

1. Let "In" be the identity matrix (n x n).

2. Let "e1" be an elementary row operation such that, applied to In, we obtain the elementary matrix E1.

3. Let "e2" be another elementary row operation such that, applied to E1, we obtain the elementary matrix E2.

4. Let A be an n x n matrix.

Then, if we apply "e1" to A, we obtain the matrix E1A, and if we apply "e2" to this last one, we obtain the matrix E2E1A.

Thanks for confirmation (or correction) of this proposition.

 

[mathwonk]

i say yes, IF when you multiply matrices, you do as i do, namely dot the ith row of the elft matrix with the jth column of the right matrix, to get the entry in the ith row and jth column of the answer.

just check your formula on a simple 2 by 2 matrix, say by adding the first row to the second row. note this is also the same as adding the second column to the first column.

take the resulting matrix and multiply it on both right and left with an arbitrary matrix with entries a,b,c,d to see what happens.

or just reason it out: the columns of a mtrix, under my conventions, contain the images of the standard basis vectors via your map. hence adding the first row to the second in the identity amtrix defiens the mapo that sends the first absis vector to the sum of the first and second basis vectors.

multiplying this by the matrix A, in the form EA, them means applying that map to the columns of A.

now the columns of A have their first basis vector component measured by their first row entry, so the first row entries should geta dded to the second row entries, ... so it seems ok.

again all is dependent on your definition of amtrix multiplication, and some books, e.g. herstein, do it differently. so questions like yours cannot be answered until their meaning ahs been made clear.

always try to go back to the meaning of the operation, not just the rule for symbol pushing.

 

[mathwonk]

if your course has not mentioned the maps defiend by a matrix but is only concerned with manipulations of matrices, try to read further and get some conceptual understanding of these operations.


i.e. a linear map from R^n to R^n is a map f such that for all vectors v,w, in R^n, we have f(v+w) = f(v) + f(w). and for all scalars t, we also have f(tv) = tf(v).

since this means you know the value of f on any vector obtained bya dding and scalar multiplication from vectors whose valoues you already know, it suffices to know the value of such an f on only the standard vectors (1,,0,...,), (0,1,0,...,0),...(0,0,...,0,1).

those values are the columns of the matrix.

then to evaluate f on a vector v = (a1,...,an) you write v as a column and multiply it by your matrix with the matrix on the left, using my rules for matrix mult above.


sinve many natural maps are linear, such as differentiation, then one can use matrices to make calculations about important subjects like calculus and differential equations.

try not to get stuck in a rut of just manipulating matrices for the sake of making computations.

 

[Castilla]

Thanks for answering, Mathwonk. In this moment the boss is patroling,
so I can't read in peace your post, but I will do so in some minutes.

 

[mathwonk]

moral: learn what a linear transformation is, and how matrices represent them.

 

[mathwonk]

you seem to be working on the plantation where the inmates are not allowed to learn.

 

[Castilla]

Something's not working well here (could be my brain, yes).

For reasons beyond my control, I can't use latex anymore. The post will look awful.

My first elementary row operation will be to triple (excuse bad english) the entries of the first row of the 2 x 2 identity matrix. I obtain matrix E1:

3 0
0 1

Now let's call A to this other 2 x 2 matrix:

3 10
2 7

If I apply my said row operation to A, I obtain

9 30
2 7

and it is easy to verify that this new matrix is equal to matrix E1.A.

Now, my second elementary row operation (appliable to matrix E1) will be to double the entries of its second row. I obtain (and denote E2):

3 0
0 2.

If I have understood the proposition posted in my first post (and blessed by Mathwonk), if I apply the second operation to matrix E1.A I should obtain a result equal to E2.E1.A. Let's see.

Doubling the entries of the second row of matrix E1.A, I obtain:
9 30
4 14.

But multiplying E2 by E1A does not give me that matrix. (Remember that E2 =
3 0
0 2

and E1A is
9 30
2 7 ).

I know this post looks awful but, well, maybe you can help with two or three words?

 

[shmoe]

Try changing E2 to:

1 0
0 2

Multiplying E1A by this guy will multiply it's last row by 2.

Notice E2E1 will then be

3 0
0 2

Multiply A by this and what happens (joy to the associative property of matrix multiplication).

I hope you feel more comfortable with these awful looking matrices since you now you're not the only one producing them.

 

[Castilla]

That works. Thank for your help, Shmoe.