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Problem 62: Determine whether it is a subspace of the given vector space $\V$
$V = M_n(\R)$, and $S$ is the subset of all $n\times n$ lower triangular matrices

Problem 62: Determine whether it is a subspace of the given vector space $\V$
$V = M_n(\R)$, and $S$ is the subset of all $n\times n$ lower triangular matrices.

Solution:
$S = \{A = \left[a_{ij} \right] \in M_n(\R) : a_{ij} = 0$ whenever $i < j\}$.
Note that $S \ne \varnothing $ ; since $0_n \in S$.
Now let $A = \left[a_{ij}\right]$ and $B = \left[b_{ij}\right]$ be lower triangular matrices.
Then $a_{ij} = 0$ and $b_{ij} = 0$ whenever $i < j$.
Then $a_{ij} + b_{ij} = 0$ and $ca_{ij} = 0$ whenever $i < j$.
Hence $A + B = \left[a_{ij} + b_{ij}\right]$ and $cA = \left[ca_{ij}\right]$ are also lower triangular matrices.
Therefore $S$ is closed under addition and scalar multiplication.
Consequently, $S$ is a subspace of $M_n(\R)$.

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