Why Is Linear Algebra So Challenging to Understand?

[nietzsche]

I had a quiz last Thursday about matrices, and I thought I knew what I was doing, but when I got there, nothing made sense to me. I'm pretty frustrated because I'm having a lot of trouble grasping any of the concepts I'm learning in linear algebra, while I'm doing fine in my "regular" math courses. Linear algebra just seems so abstract and...wrong...

Homework Statement

Write A as a product of elementary matrices. (Hint: First, write the inverse as a product of elementary matrices.)

<br /> A =<br /> \left[<br /> \begin{array}{cc}<br /> 1 &amp; 2 \\<br /> 3 &amp; 4<br /> \end{array}<br /> \right]<br />

Homework Equations

The Attempt at a Solution

I just don't understand why any of this works. I know how to find the inverse using Gauss-Jordan elimination. I wrote the partitioned matrix with A on the left and the 2x2 identity on the right. Then I performed the following row operations:

R2 - 3R1
R1 + R2
(-1/2)R2

and I got the following partitioned matrix:

<br /> \left[<br /> \begin{array}{cc|cc}<br /> 1 &amp; 0 &amp; -2 &amp; 1 \\<br /> 0 &amp; 1 &amp; \frac{3}{2} &amp; -\frac{1}{2}<br /> \end{array}<br /> \right]<br />

I simply don't understand the concept of how a matrix can be a product of elementary matrices. I don't see how a matrix can correspond to an elementary row operation. Please help me understand! I'm seriously beginning to go a bit loopy. Thanks.

 

[n!kofeyn]

Maybe http://www.math.unm.edu/~loring/links/linear_s08/elementary.pdf" will help. It goes through the process with a 3x3 matrix. You basically find your inverse matrix, keep track of the row operations, and then you translate the reverse of these row operations into elementary matrices, which gives you your product of elementary matrices.

Note that you can only do this if the matrix is invertible.

 

[HallsofIvy]

An "elementary matrix" is a matrix derived from the identity matrix by a single rwo operation. The crucial point is that multiplying any matrix, on the left, by an elementary matrix does that row operation to the matrix.

For example, if I "add two times the first row to the third" with the identity matrix I get
\begin{bmatrix}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 2 &amp; 0 &amp; 1\end{bmatrix}

If I multiply any matrix by that
\begin{bmatrix}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 2 &amp; 0 &amp; 1\end{bmatrix}\begin{bmatrix} a &amp; b &amp; c \\ d &amp; e &amp; f \\ g &amp; h &amp; i\end{bmatrix}= \begin{bmatrix}a &amp; b &amp; c \\ d &amp; e &amp; f \\ 2a+ g &amp; 2b+ h &amp; 2c+ i\end{bmatrix}

Now, suppose that, starting with matrix A, you do row operations x, y, and z (first x, then y, then z) to reduce it to the identity matrix. Writing X, Y, and Z as the corresponding elementary matrix, that says that Z(Y(XA))= I. But matrix multiplication is "associative". That is the same as (ZYX)(A)= I which says precisely that ZYX is the inverse matrix to A. Of course, ZYX is the same as ZYXI which is the same as performing row operations x, y, and z on I.

 

[nietzsche]

wonderful, thanks very much to both of you