# Linear Algebra - Domain or co-domain for matrix A=[1,2;2,1;1,1] ?

#### KindofSlow

##### Junior Member
Hello,
This definition:
Codomain The codomain of a linear transformation is the vector space which contains the vectors resulting from the transformation's action. Thus, if T(v) = w, then v is a vector in the domain and w is a vector in the range, which in turn is contained in the codomain.
Examples:
• The codomain of the transformation T:R3→R5 is R5
• The matrix A=[1,2;2,1;1,1] (three rows and two columns) induces a linear map from R2 to R3, with domain R3
My question is regarding the very last statement:
• The matrix A=[1,2;2,1;1,1] (three rows and two columns) induces a linear map from R2 to R3, with domain R3
I think the domain is R2 since there are 2 columns in A and since the linear map is from R2, so v has 2 elements in Av=w
Since we go from Domain to Range, v to w, and R2 to R3, I think Domain should be R2, and not R3.
I think the range of A is a subspace of R3.
I think the codomain of A is all of R3.
Any assistance with anything I am misunderstanding or anything I have wrong will be greatly appreciated.
Thank you

#### Dr.Peterson

##### Elite Member
I think it's a typo. They clearly mean to say that the codomain is R3, not the domain. After all, that's the word they're demonstrating, so it should be used in the example! Possibly a proofreader hadn't heard of codomain and "corrected" it to domain.

The codomain is the set B in the notation f:A→B, which contains the range. That is, it is the set you are mapping into.

#### KindofSlow

##### Junior Member
Makes total sense. Thank you.