# Linear algebra - Linear transformations

#### pka

##### Elite Member
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
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
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)

• Jomo