In order to determine whether or not a transformation is both linear and isomorphic we must check to see if it meets the following criteria:

In order to determine linearity, usually the following two properties are sufficient:

T(f+g)=T(f)+T(g)

T(kf)=kT(f)

In order to determine Isomorphism, we must check to see if the kernel of the transform is 0. Which also means for a given transformation T from V to W, the image is W, fulfilling the rest of the requirements for isomorphism. Thus we can just check to see if the kernel is 0 or non-zero.

Example:

T(M)=M |1 2|

|3 6|

M=|a b|

|c d|

|a+3b 2a+6b|

|c+3d 2c+6d|

=

|0 0|

|0 0|

We can then separate this into solving for a,b which will be replicated for c,d

|1 3 0|

|2 6 0|

a = -3b

This makes M =

|-3b b|

|-3d d|

Thus the kernel is nonzero, and so this is not an isomorphic transformation.

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