Which of the subsets of PSKR^{3×3}ZSK are subspaces of PSKR^{3×3}ZSK? NSK a)

Inyalan0 2021-12-11 Answered
Which of the subsets of R3×3 are subspaces of R3×3?
a) The invertible 3×3 matrices
b) The 3×3 matrices whose entries are all integers
c) The 3×3 matrices with all zeros in the third row
d) The non-invertible 3×3 matrices
e) The diagonal 3×3 matrices
f) The symmetric 3×3 matrices
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Expert Answer

Steve Hirano
Answered 2021-12-12 Author has 34 answers
encolatgehu
Answered 2021-12-13 Author has 27 answers
If the entries are integers and the linear combination includes π as a coefficient, then the entries in the resulting matrix will usually not be integers. Just try it in some simple cases where there are just two terms, with coefficients π and 0.
As for (c), a sum of non-invertible matrices is often invertible.
(e) and (f) are subspaces.
It is redundant to say they include 0 if they're closed under linear combinations, because there is a linear combination in which all coefficients are zero.
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