How to find the linear transformation associated with a given matrix? It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation T:V→W can be represented as a matrix, but since this is an isomorphism between L(V,W) and Mm×n where the latter represents the space m×n matrices on the same field in which are defined respectively vector spaces.

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How to find the linear transformation associated with a given matrix?  It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation T:V→W can be represented as a matrix, but since this is an isomorphism between L(V,W) and Mm×n where the latter represents the space m×n matrices on the same field in which are defined respectively vector spaces.

coolbananas03ok · Accepted Answer

Step 1  I am not sure what you expect, say T is your linear transform, and A represents it in the basis for  V={a1,⋯,an} and W={b1,⋯,bm}  Then we say Taj=∑iAijbi.  This should completely define what T &quot;does&quot;. Typically, one defines the action of T on the basis vectors ai in order to determine what T &quot;looks like&quot;. Otherwise, one can not write down much about T. Please clarify what type of answer you are expecting if this is not sufficient.

Howard Gallagher · Accepted Answer

Step 1  The columns of the matrix tell us where the basis vectors of the domain are mapped, in terms of the basis vectors of the codomain. Since every vector in the domainis a linear combination of the basis vectors (in a unique way), we can extrapolate, in a sense, the image of any given vector. Let A be an m×n matrix (with coefficients in a field F) with columns A1,⋯,An  Let V be an n-dimensional F-vector space, and W an m-dimensional F-vector space, with ordered bases (v1,⋯,vn) and (w1,⋯,wm), respectively. Finally, let T be the linear transformation associated with A, and let v∈V with v=c1v1+⋯+cnvn  (remember, this expression for v as a linear combination of basis vectors is unique). Then  T(v)=T(c1v1+⋯+cnvn)=c1T(v1)+…+cnT(vn)=c1A1+⋯+cnAn So, this is how the matrix lets us calculate the image of any vector. Notice that the expression on the right is just the matrix A multiplied by the vector [c1,c2,⋯,cn]T  For example, if we let V=W=BR2 (considered as BR− vector spaces) with the standard basis, let  A=[12 23]  And let T be the linear transformation associated with A, and let  v=[1, 5]T. Then  T(v)=1⋅T(e1)+5⋅T(e2)=[1,2]T+5[2,3]T=[11,17]T=A⋅v

How to find the linear transformation associated with a given matrix? It is well known that give

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