Linear Transformation is a term used often in physics. For example, we are told here that a tensor is a generalization of a linear transformation. I've never actually learned what they are, though. Could someone explain?
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3Do you know what a vector space is? – TimRias Feb 13 '23 at 15:06
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Yes, I do. Why? – Feb 13 '23 at 15:08
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A linear transformation $A:V\to W$ is a map between vector spaces $V$ and $W$ such that for any two vectors $v_1, v_2 \in V$,
$$A(v_1+v_2) = A(v_1) + A(v_2),$$
and for any scalar $\lambda$,
$$ A(\lambda v_1) = \lambda A(v_1).$$
In other words a linear transformation is a map between vector spaces that respects the linear structure of both vector spaces.
TimRias
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@user45664 it follows only for integral values of $\lambda$, not all $\lambda$ – Aditya_math Feb 13 '23 at 21:50
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1@user45664: I think the second equation follows from the first --- Assuming that we're dealing with scalars that are real numbers, then the second equation for rational numbers $\lambda$ follows from the first equation, but not for all real numbers. In particular, can you prove $A\left(\sqrt{2}v_1\right) = \sqrt{2}A(v_1)$ using only the first equation? – Dave L. Renfro Feb 13 '23 at 21:52
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1Regarding my previous comment, see Cauchy's functional equation. Functions that satisfy the first equation but not the second equation are extremely pathological. See Graph of discontinuous additive function is dense in $\mathbb R^2$ and Overview of basic facts about Cauchy functional equation. FYI, having a dense graph can actually be achieved by much less pathological functions. – Dave L. Renfro Feb 13 '23 at 22:09