Answer:
A) The Matricial formula for T is:
[tex]T(V)=\left[\begin{array}{ccc}57&73&81\\93&101&113\\29&34&38\end{array}\right] V[/tex]
B) The type of phones produced are:
[tex]\left[\begin{array}{ccc}J8\\J8+\\J9\end{array}\right] =\left[\begin{array}{ccc}7\\8\\12\end{array}\right][/tex]
Step-by-step explanation:
Let us define the transformation where:
[tex]V=(j8, j8+, j9)=(A, B, C)[/tex]
[tex]T(V)=(T_{Labor}, T_{materials},T_{Overhead})[/tex]
In this case:
[tex]T_{Labor}=57A+73B+81C\\T_{materials}=93A+101B+113C\\T_{Overhead}=29A+34B+38C[/tex]
If we define a base for this transformation for the space V/:
V∈V/ and W' a base of the vectorial space V/:
[tex]W'=\{(1,0,0)^T(0,1,0)^T(0,0,1)^T\}[/tex]
T(V)∈W/ and W'' a base of the vectorial space W/:
[tex]W''=\{(1,0,0)^T(0,1,0)^T(0,0,1)^T\}[/tex]
The transformation T defined in the base W' is:
[tex]T(w'_1)=T(1,0,0)=(57, 93,29)^T\\T(w'_2)=T(0,1,0)=(73,101,34)^T\\T(w'_3)=T(0,0,1)=(81, 113,38)^T[/tex]
Therefore the transformation's Matrix in the bases W' and W'' is defined as:
[tex]T(V)_{W''}=\left[\begin{array}{ccc}57&73&81\\93&101&113\\29&34&38\end{array}\right]_{W'W''} V_{W'}[/tex]
In this case, the bases W' and W'' are the canonical ones. Therefore we can simply write the formula as:
[tex]T(V)=\left[\begin{array}{ccc}57&73&81\\93&101&113\\29&34&38\end{array}\right]V[/tex]
To determine T⁻¹ we need to know if there is some element in the nullspace.
If there is no element in the nullspace, this transformation is an isomorphism, therefore there is a single solution for T⁻¹. To see quickly if there are elements in the nullspace, we can use the determinant of the transformation's matrix. If this determinant is different than 0, there aren't any elements in the nullspace.
[tex]Det|\left[\begin{array}{ccc}57&73&81\\93&101&113\\29&34&38\end{array}\right]|=-116\neq 0 \leftrightarrow Null([T]=\{0\}[/tex]
we can calculate the inverse matrix of [T] knowing that it exists. In this case:
[tex][T]^{-1}=\left[\begin{array}{ccc}1/29&5/29&-17/29\\257/116&183/116&-273/29\\-233/116&-179/116&258/29\end{array}\right][/tex]
Therefore the inverse transformation of T is defined as:
[tex]T^{-1}(W)=[T]^{-1}W=[T]^{-1}(T(V_?))[/tex]
In B), we know that T(V?) is (1955,2815,931). Therefore if we replace it in the equation above:
[tex][T]^{-1}((1955,2815,931)^T)=\left[\begin{array}{ccc}1/29&5/29&-17/29\\257/116&183/116&-273/29\\-233/116&-179/116&258/29\end{array}\right] \cdot \left[\begin{array}{ccc}1955\\2815\\931\end{array}\right] =\left[\begin{array}{ccc}7\\8\\12\end{array}\right][/tex]
Therefore the type of phones produced are:
[tex]\left[\begin{array}{ccc}J8\\J8+\\J9\end{array}\right] =\left[\begin{array}{ccc}7\\8\\12\end{array}\right][/tex]
