Show that every linear transformation is the sum of two invertible linear transformations .

Printable View

- May 12th 2010, 12:31 PMBruno J.[SOLVED] Linear transformations
Show that every linear transformation is the sum of two invertible linear transformations .

- May 12th 2010, 01:05 PMNonCommAlg
- May 12th 2010, 11:22 PMBruno J.
Good! Minced meat for NCA! (Nod)

Here's a topological proof I found. It's much less direct but maybe the idea can be used elsewhere! Let's equip with the Euclidean metric. Note that is dense in . Let . Note that is not dense in .

If is invertible, then the theorem is trivial with .

Therefore suppose . Let . It's clear that is dense in since is dense in . If , then and therefore is dense in which is false. Therefore . - May 12th 2010, 11:35 PMDrexel28
- May 12th 2010, 11:42 PMBruno J.
- May 12th 2010, 11:44 PMDrexel28
- May 12th 2010, 11:47 PMBruno J.
- May 12th 2010, 11:48 PMBruno J.
- May 12th 2010, 11:50 PMDrexel28