Can anyone help me with vector spaces?
Let ( F ) be a fixed ( 3 \times 2 ) matrix, and let ( H ) be the set of all ( 2 \times 4 ) matrices ( A ) in ( M ) such that ( FA = 0 ) (where ( 0 ) is the zero matrix in ( N ), a ( 3 \times 4 ) matrix). I would like to determine whether ( H ) is a subspace of ( M ) (the space of ( 2 \times 4 ) matrices). Thank you in advance for your assistance.
1 Answers
Jan 12, 2025
If H is a subspace, then it has to fit all three conditions:
1) A contains the zero matrix.
2) If u and v are in A, then so is u+v.
3) If u is in A and k is a scalar, then ku is in A too.
#1 is true because if we fill H up with all zeroes, then obviously FH=0 should still hold true.
For #2, let’s assume u and v are in A. Then Fu = 0 and Fv = 0. That means Fu + Fv = 0, and F(u+v)=0 by the distributive property.
For #3, assume u is in A. Then for scalar k, Fu=0; Fu*k = 0, and so F(uk) = 0
So let’s assume u and
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