Question

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. [Hint: Given u, v in \({\mathbb{R}^n}\), let \[{\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\]. Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \[T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \]. Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).]

Short answer

It is proved that Sis a linear transformation.

Step-by-step solution

Show that S preserves sums

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be a linear transformation and Abe the standard matrix for T. Then, according totheorem 9, Tis invertible if and only if Ais an invertible matrix. The linear transformation S, given by \[S\left( x \right) = {A^{ - 1}}{\mathop{\rm x}\nolimits} \], is the unique function satisfying the equations

1. \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \), for all x in \({\mathbb{R}^n}\), and
2. \(T\left( {S\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \), for all x in \({\mathbb{R}^n}\).

It is given that u, and v are in \({\mathbb{R}^n}\).

If \[x = S\left( {\mathop{\rm u}\nolimits} \right)\] and \[y = S\left( {\mathop{\rm v}\nolimits} \right)\], then \[T\left( x \right) = T\left( {S\left( {\mathop{\rm u}\nolimits} \right)} \right) = {\mathop{\rm u}\nolimits} \] and \[T\left( y \right) = T\left( {S\left( {\mathop{\rm v}\nolimits} \right)} \right) = {\mathop{\rm v}\nolimits} \] from equation (2). Thus,

\[\begin{array}{c}S\left( {{\mathop{\rm u}\nolimits} + {\mathop{\rm v}\nolimits} } \right) = S\left( {T\left( x \right) + T\left( y \right)} \right)\\ = S\left( {T\left( {x + y} \right)} \right)\,\,\,\,{\rm{ (}}{\mathop{\rm Since}\nolimits} \,\,T\,\,{\mathop{\rm is}\nolimits} \,\,{\mathop{\rm linear}\nolimits} )\\ = x + y\,\,\,\,\,{\rm{ (}}By\,\,equation\,\,\left( 1 \right))\\ = S\left( {\mathop{\rm u}\nolimits} \right) + S\left( {\mathop{\rm v}\nolimits} \right).\end{array}\]

Therefore, Spreserves sums.

Show that S is a linear transformation

For any scalar r,

\[\begin{array}{c}S\left( {r{\mathop{\rm u}\nolimits} } \right) = S\left( {rT\left( {\mathop{\rm x}\nolimits} \right)} \right)\\ = S\left( {T\left( {r{\mathop{\rm x}\nolimits} } \right)} \right)\,\,\,\,{\rm{ (}}{\mathop{\rm Since}\nolimits} \,\,T\,\,is\,\,linear)\\ = r{\mathop{\rm x}\nolimits} \,\,\,{\rm{ (}}\,{\mathop{\rm By}\nolimits} \,\,equation\left( 1 \right))\\ = rS\left( {\mathop{\rm u}\nolimits} \right).\end{array}\]

Spreserves the scalar multiples, and hence,itis a linear transformation.

Thus, it is proved that Sis a linear transformation.