Linear algebra - Linear transformations
- Thread starter akleron
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pka
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You must tell us what the notation \(\displaystyle \large\left[T\right]_C^B\) means. It is not a standard far as I know.
Since a linear change of basis can always be performed by multiplication by an appropriate matrix both suppositions are almost trivial to prove.
\(\displaystyle \text{Let $T$ be the matrix $T$ expressed in the identity matrix basis.}\\
\text{Then for some matrix $B,~[T]^B = BT$ and likewise $[T]^B_C = CBT$}
\\
\text{so $[T+S]^B_C = CB(T+S) = CBT +CBS = [T]^B_C + [ S]^B_C $ }\)
you can do (ii)
\(\displaystyle \text{Let $T$ be the matrix $T$ expressed in the identity matrix basis.}\\
\text{Then for some matrix $B,~[T]^B = BT$ and likewise $[T]^B_C = CBT$}
\\
\text{so $[T+S]^B_C = CB(T+S) = CBT +CBS = [T]^B_C + [ S]^B_C $ }\)
you can do (ii)