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

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- May 12th 2010, 01:31 PMBruno J.[SOLVED] Linear transformations
Show that every linear transformation is the sum of two invertible linear transformations .

- May 12th 2010, 02:05 PMNonCommAlg
- May 13th 2010, 12:22 AMBruno 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 13th 2010, 12:35 AMDrexel28
- May 13th 2010, 12:42 AMBruno J.
- May 13th 2010, 12:44 AMDrexel28
- May 13th 2010, 12:47 AMBruno J.
- May 13th 2010, 12:48 AMBruno J.
- May 13th 2010, 12:50 AMDrexel28