Question

Let M33(R) be the set of 3 x 3 matrices over real numbers and M33(R) is a vector space. Let W be the set of 3 x3 matrices A in M33(R) with A =-A^T. i.e., W = {A is an element of M33(R) : A = -A^T}. (a). Prove that W is a subspace of M33(R). (b). From (a) we can see that W is a vector space itself. Find the bases for V? (Justify) (c). Find the dimension of W.​

Answer 1

Answer:

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Step-by-step explanation:

Since M 3x3( R) is a Euclidean vector space (isomorphic to R 9), all that is required to establish that S 3x3( R) is a subspace is to show that it is closed under addition and scalar multiplication. If A = A T and B = B T, then ( A + B) T = A T + B T = A + B, so A + B is symmetric; thus, S 3x3( R) is closed under addition. Furthermore, if A is symmetric, then ( kA) T = kA T = kA, so kA is symmetric, showing that S 3x3( R) is also closed under scalar multiplication.