lybrain354
29.07.2019 •
Mathematics
The trace of a square n×n matrix a=(aij) is the sum a11+a22+⋯+ann of the entries on its main diagonal. let v be the vector space of all 2×2 matrices with real entries. let h be the set of all 2×2 matrices with real entries that have trace 0. is h a subspace of the vector space v?
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Ответ:
Let and be two vectors in .
is a subspace of if (1) and (2) for any scalar , we have .
(1) True;
so .
(2) Also true, since
Therefore is a subspace of .
Ответ:
Yes, H is a subspace of V
Step-by-step explanation:
We know that V is the space of all the 2x2 matrices with real entries.
H is the set of all 2x2 matrices with real entries that have trace equal to 0.
Obviusly the matrices that are in the space H also belong in the space V (because in H you have some selected matrices and in V you have all of them). The thing we need to prove is if H is an actual subspace.
Suppose we have two matrices that belong to H, A and B.
We must see that:
1) if A and B ∈ H, then (A + B)∈H
2) for a scalar number k, k*A ∈ H
lets write this as:
where a1 + a4 = 0 = b1 + b4
then:
the trace is:
a1 + b1 - (a4 + b4) = (a1 - a4) + (b1 - b4) = 0
then the trace is nule, and (A + B) ∈ H
and:
the trace is:
k*a1 - k*a4 = k(a1 - a4) = 0
so kA ∈ H
then H is a subspace of V
Ответ:
33.51
Step-by-step explanation:
hope this helps