S is a vector subspace of vector space V? (matrix in complex) V=M_2(CC) , K=CC and S={((a_11,a_12),(a_21,a_22)) in V :a_(ij)=bar(a_(ji)) text( for )i,j=1,2}

daniellex0x0xto

daniellex0x0xto

Answered question

2022-08-12

S is a vector subspace of vector space V? (matrix in complex)
V = M 2 ( C ) , K=C and
S = { ( a 11 a 12 a 21 a 22 ) V : a i j = a j i ¯  for  i , j = 1 , 2 } .
I'd like to understand better how to work with complex and matrix to check this question. I know there are 3 rules, but id like how to apply in cases like that. Thanks in advance, ill be glad for explanations.

Answer & Explanation

Alejandra Blackwell

Alejandra Blackwell

Beginner2022-08-13Added 14 answers

For i { 1 , 2 }, a i i ¯ = a i i implies that a i i is a real number, so
S = { ( x z z ¯ y ) V : x , y R  and  z C } .
Now, is the sum of two matrices in S a matrix in S? and the scalar multiplication of a matrix in S with a complex number?

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