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please give the clear steps of any questions and writing or typing clearly please. it's better to type the answers. many thanks!

Question 3. Lat U be the set of symmetrie matrices in Ma(C that is,the set of 3 x3 K n complex matrices Show that U is a vector space over C in its own right, with the usual rules for matrix addition and multiplication by scalars. What is the dimension of U? Hint: For the second part, you need to find a basis for U. The matrices EI, E2 and E3s from the standard basis for M3x3(C) are already symmetric, but you need to find a way of pairing u p the remaining basis vectors Eij for i

Explanation / Answer

Let A and B be 2 arbitrary elements of U and let be an arbitrary complex scalar. Then AT = A and BT = B as both A and B are symmetric matrices. Further, (A+B)T = AT +BT = A+B so that A+B U. Thus, U is closed under vector addition. Also, (A)T = AT = A so that A U. Thus, U is closed under scalar multiplication. Further, the 3x3 zero matrix, being a complex symmetric matrix ( 0= 0+0i) U. Hence U is a vector space.

The dimension of U is 6 as the standard basis for U is {E1,E2,E3,E4,E5,E6} where E1 has 1 in its a11 position, the rest of the entries being 0, E2 has 1 in its a22 position, the rest of the entries being 0, E3 has 1 in its a33 position, the rest of the entries being 0, E4 has 1 in its a12 and a21 positions, the rest of the entries being 0, E5 has 1 in its a13 and a31 positions, the rest of the entries being 0 and E6 has 1 in its a23 and a32 positions, the rest of the entries being 0.

Note:

1 is also a complex number as 1 = 1+0i. Also 1*(a+ib) = a+ib.

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