Theorem: *A subspace*

-----------------

Let W_n be the subset of all M_(n x n)(

To show that W_n is a subspace of M_(n x n)(**R**) it is sufficient to demonstrate that the three conditions of the theorem are satisfied.

S1) In M_(n x n)(**R**) there exists a matrix Z:{z_ij = 0, for all 1 < i,j < n} (ie. a matrix such that for every row i or column j consists of the zero vector).

Since z_ij = z_ji it is a symmetric matrix and must also be in W_n.

To satisfy S2,3) it is necessary to show the property of transpose as it relates to addition and scalar multiplication of matrices.

Let A, B be matrices in M_(n x n)(**R**), let a,b be scalars in **R**. Then let O = aA + bB and let P = a(A^t) + b(B^t). Then:

O_ij = a(A_ij) + b(B_ij) = P_ji => P_ij = O_ji

S2) If A,B are in W_n, with A = A^t and B = B^t then:

(A + B)^t = A^t + B^t = A + B => A + B is in W_n.

Hence W_n is closed under addition.

S3) If A is in W_n then for all a in **R**:

(aA) = a(A^t) = aA => aA is in W_n.

Hence W_n is closed under scalar multiplication.

Thus by theorem we see that W_n ≤ M_(n x n)(**R**).** []**

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