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Upper trianglar matrix is a subspace of mxn matrices

  1. Oct 9, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that the upper triangular matrices form a subspace of ##\mathbb{M}_{m \times n}## over a field ##\mathbb{F}##

    2. Relevant equations


    3. The attempt at a solution
    We can prove this entrywise.

    1) Obviously the zero matrix is an upper triangular matrix, because it satisfies the property that whenever ##i > j##, ##A_{ij} = 0##.

    2) There are two cases; when ##i \le j## and when ##i > j##.
    When ##i \le j## we have that ##(A + B)_{ij} = A_{ij} + B_{ij}##. When ##i > j## we have that ##(A + B)_{ij} = A_{ij} + B_{ij} = 0 + 0 = 0##. Therefore, ##A + B## is still an upper triangular matrix.

    3) When ##i \le j## we have that ##(cA)_{ij} = c(A_{ij})##. When ##i > j##, we have that ##(cA)_{ij} = c(A_{ij}) = c(0) = 0##. Therefore, ##cA## is still an upper triangular matrix.

    Is this argument enough to show that the upper triangular matrices form a subspace of mxn matrices over a field F?
     
  2. jcsd
  3. Oct 9, 2016 #2

    andrewkirk

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