Linear Transformation

up:: Vector Space

Let $(V,+,\cdot ),(W,{+}^{\prime },{\cdot }^{\prime })$ be vector spaces over the same Field $\mathbb{K}$. A transformation $T:V\to W$ is said to be a linear transformation if

$$\forall u,v\in V;\forall \lambda \in \mathbb{K}:T(u+\lambda v)=T(u)+\lambda T(v)$$

Thus, it preserves the vector space operations from $V$ to $W$.

Properties

• If $W=V$, we say that $T:V\to V$ is an Endomorphism.
• A linear transformation is injective iff its kernel is trivial
• A linear transformation’s image of a basis spans its image
• A linear transformation is injective iff it preserves linear independence
• A linear transformation between spaces of equal dimension is injective iff it’s surjective iff it’s an isomorphism

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References

• Notas sobre Álgebra Linear