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Let me give another answer to 1).

In general, given a linear mapping $$\phi \colon E \to F$$ it extends uniquely to a homomorphism $$T(\phi) \colon T(E) \to T(F).$$ The proof can be made coordinate-free, in fact it follows from the universal property of $T(E)$ applied to the map $$\eta \colon E \to T(F),$$ where $\eta=i \circ \phi$ and $i \colon F \to T(F)$ is the natural embedding.

By construction it follows

$$T(\phi)(x_1 \otimes \ldots \otimes x_p)=\phi x_1 \otimes \ldots \otimes \phi x_p,$$x_p.$$If \psi \colon F \to G is another linear map one obtains$$T(\psi \circ \phi)=T(\psi) \circ T(\phi),$$hence T(\phi) is injective [risp. surjective] whenever \phi is injective [resp. surjective]. For more details, see for instance [Greub, Multilinear Algebra, Chapter III]. 2 added 116 characters in body; deleted 4 characters in body Answer Let me give another answer to 1). In general, given a linear mapping$$\phi \colon E \to F$$it extends uniquely to a homomorphism$$T(\phi) \colon T(E) \to T(F).$$The proof can be made coordinate-free, in fact it follows from the universal property of T(E) applied to the map$$\eta \colon E \to T(F),$$where \eta=i \circ \phi and i \colon F \to T(F) is the natural embedding. Moreover, one proves easily that By construction it follows$$T(\phi)(x_1 \otimes \ldots \otimes x_p)=\phi x_1 \otimes \ldots \otimes \phi x_p,$$hence T(\phi) is injective [risp. surjective] whenever \phi is injective [resp. surjective]. For more details, see for instance [Greub, Multilinear Algebra, Chapter III]. 1 Answer to 1). In general, given a linear mapping$$\phi \colon E \to F$$it extends uniquely to a homomorphism$$T(\phi) \colon T(E) \to T(F).$$The proof can be made coordinate-free, in fact it follows from the universal property of T(E) applied to the map$$\eta \colon E \to T(F), where $\eta=i \circ \phi$ and $i \colon F \to T(F)$ is the natural embedding.

Moreover, one proves easily that $T(\phi)$ is injective [risp. surjective] whenever $\phi$ is injective [resp. surjective].

For more details, see for instance [Greub, Multilinear Algebra, Chapter III].