Linear algebra - Linear transformations

akleron

New member
Joined
Dec 28, 2019
Messages
32
I need to show to following two statements:
Tbh, I've got no idea how to start.
Would love some help..
Thanks !!

1579039017506.png
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
9,174
I need to show to following two statements:
Tbh, I've got no idea how to start.
Would love some help..
Thanks !!

View attachment 16076
You must tell us what the notation \(\displaystyle \large\left[T\right]_C^B\) means. It is not a standard far as I know.
 

akleron

New member
Joined
Dec 28, 2019
Messages
32
You must tell us what the notation \(\displaystyle \large\left[T\right]_C^B\) means. It is not a standard far as I know.
It means apply the function T on v (when v belongs to the span of base B) and than write the coordinate vector of the result over base C
does it make sense ?
 

Romsek

Full Member
Joined
Nov 16, 2013
Messages
678
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)
 
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