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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.]
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