1. Let A be a 3 by 4 matrix.

A =

 1 -4 8 1 0 2 -1 3 0 0 0 5

My textbook says that in order for the columns of A to span Rm, A must have a pivot position in every row. We see that it does.

My problem is that I thought that the matrix

 1 -4 8 1 0 2 -1 3 0 0 0 0 0 0 0 5

meant the exact same thing as the matrix A. But then it would not span Rm because there is a row with no pivot position. What am I not understanding about matrices?

2.

3. Compare determinants of your 2 matrices. Are they the same?

4. The first matrix maps R4 into R3, the second maps R4 into R4. However, the third coordinate of the second map is 0, so there is a 1-1 correspondence between the images of the two maps.

5. Originally Posted by Guitarist
Compare determinants of your 2 matrices. Are they the same?
We haven't covered "determinants" yet.

Another reason why I don't understand the difference between the two matrices is that the solution set would seem to be the exact same for Ax = b.

6. Okay, thanks for the help Guitarist and mathman!

7. 1. If any row or column of a matrix is all zeros, then the determinant is zero - as you can easily check

2. If the determinant of a matrix is zero, then the row (or column) vectors are not linearly independent, and therefore cannot be a spanning set

 Bookmarks
##### Bookmarks
 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Terms of Use Agreement