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 2 years ago
how do you show the set V of all 2 x 2 upper triangular matrices is a subspace of M22?
 2 years ago
how do you show the set V of all 2 x 2 upper triangular matrices is a subspace of M22?

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satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0verify all the conditions for the definition of a subspace, which is really the point of this exercise, to write them down

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0for example, the identity matrix has to be included in the subspace, and \[\begin{pmatrix} 1& 0 \\ 0 & 1 \end{pmatrix}\] is upper triangular, so it is in the subspace

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0you have to show it is closed under addition check that \[\begin{pmatrix} a & b \\ c & 0 \end{pmatrix}+\begin{pmatrix} e& f \\ g & 0 \end{pmatrix}\] is upper triangular, which it is etc

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0check that the inverse of an upper triangular two by two matrix is also an upper triangular two by two matrix in other words, write down all the conditions necessary for it to be a subspace, then check them one by one

JamesJ
 2 years ago
Best ResponseYou've already chosen the best response.2Um, upper triangular 2x2 matrices are matrices of the form \[ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \] So some modifications to be made above. Also, the zero vector in \(M_{2 \times 2}\) is the zero matrix, not the identity matrix.

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0thanks for keeping me honest

teacherman79
 2 years ago
Best ResponseYou've already chosen the best response.0@satellite73 @JamesJ Thank you both...this site is cool...wish I knew about it 6 weeks ago...
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