/sci/ - Science & Math

[Anonymous No.10198614]

I am trying to self teach myself linear algebra and I don't understand why I am not getting the same column space for two different methods.

Consider the matrix A

1 2 -2 1
3 6 -5 4
1 2 0 3

this row reduces to this

1 2 0 3
0 0 1 1
0 0 0 0

Using the pivots, this implies that the column space for A is the span of {[1,3,1],[-2,-5,0]}

However I was taught a second method that is to take the transpose of A and row reduce, and the rows with leading ones of the transpose of A is the vectors for the span of the column space of A

transpose of A is

1 3 1
2 6 2
-2 -5 0
1 4 3

This row reduces to

1 0 -5
0 1 2
0 0 0
0 0 0

Implying that the basis for the column space of A is [1,0,-5] [0,1,2]

However, the span of {[1,3,1],[-2,-5,0]} is not the span of {[1,0,-5] [0,1,2]}

https://www.wolframalpha.com/input/?i=span%7B%5B1,3,1%5D,%5B-2,-5,0%5D%7D
https://www.wolframalpha.com/input/?i=span%7B%5B1,0,-5%5D,%5B0,1,2%5D%7D

Where is my dun goof?

[GRID-Resistant No.10198628]

don't post in this thread or you'll get GRID