Question

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

Answer 1

verify all the conditions for the definition of a subspace, which is really the point of this exercise, to write them down

Answer 2

for 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

Answer 3

you 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

Answer 4

check 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

Answer 5

Um, 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.

Answer 6

thanks for keeping me honest

Answer 7

@satellite73 @JamesJ Thank you both...this site is cool...wish I knew about it 6 weeks ago...