Question

Let \(V\) be the linear space of \(n \times n\) matrices with real entries. Define a linear transformation \(T: V \rightarrow V\) by the rule \(T(A)=\frac{1}{2}\left(A+A^{T}\right)\). [Here \(A^{T}\) is the matrix transpose of \(A .]\)
a) Verify that \(T\) is linear.
b) Describe the image of \(T\) and find it's dimension. [Try the cases \(n=2\) and \(n=3\) first.]
c) Describe the image of \(T\) and find it's dimension.
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d) Verify that the rank and nullity add up to what you would expect. [NOTE: This \(\operatorname{map} T\) is called the symmetrization operator.]