Determine if Linear

Problem

S*([a],[b],[c])=[[2*a−6*b+6*c],[a+2*b+c],[2*a+b+2*c]]

Solution

1. Identify the definition of a linear transformation. A transformation S:V→W is linear if for all vectors u,v∈V and scalars k it satisfies S*(u+v)=S(u)+S(v) and S*(k*u)=k*S(u)

2. Rewrite the transformation as a matrix-vector product. We can express the output vector as a combination of the variables a b and c

3. Extract the coefficients of a b and c to form a matrix A

S*([a],[b],[c])=a*[[2],[1],[2]]+b*[[−6],[2],[1]]+c*[[6],[1],[2]]

4. Formulate the matrix A such that S(v)=A*v

A=[[2,−6,6],[1,2,1],[2,1,2]]

5. Conclude that since the transformation can be written in the form S(v)=A*v where A is a constant matrix, the transformation satisfies the properties of additivity and homogeneity.

Final Answer

S*([a],[b],[c])=[[2,−6,6],[1,2,1],[2,1,2]]*[[a],[b],[c]]⇒Linear

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