37.3: Transformación uno a uno e inversa

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Lee la Sección 4.9 del libro de texto si no estás familiarizado con esta parte.

Definición

Se dice que una transformación\(T:U\mapsto V\) es uno a uno si cada elemento del rango es la imagen de un solo elemento en el dominio. Es decir, para dos elementos (\(x\)y\(y\)) en\(U\). \(T(x)=T(y)\)ocurre sólo cuando\(x=y\).

Teorema

Let\(T:U\mapsto V\) Ser una transformación lineal uno a uno. Si\(\{u_1,\dots,u_n\}\) es linealmente independiente en\(U\), entonces\(\{T(u_1),\dots,T(u_n)\}\) es linealmente independiente en\(V\).

Definición

Se dice que una transformación lineal\(T:U\mapsto V\) es invertible si existe una transformación\(S:V\mapsto U\), tal que\(S(T(u))=u,\qquad T(S(v))=v,\) para cualquiera\(v\) dentro\(V\) y cualquiera\(u\) en\(U\).

Pregunta 4

Si la transformación lineal\(T:U\mapsto V\) es invertible, y la dimensión de\(U\) es 2, ¿cuál es la dimensión de\(V\)? ¿Por qué?

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