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Example Question #41 : The Transpose
True or false; The set of all symmetric matrices is a subspace of all matrices.
Possible Answers:
True
False
Correct answer:
True
Explanation:
Without being too abstract, it is easy to convince oneself that this is true. We have to check the 3 criteria for a subspace.
1. Closure under vector addition
Adding together two symmetric matrices will always result in another symmetric matrix.
2. Closure under scalar multiplication.
Multiplying a symmetric matrix by a scalar will also always give you another symmetric matrix
3. The zero vector (matrix in this case) is also in the subset
Indeed the zero vector itself is a symmetric matrix.
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