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I'm pretty new to Linear Algebra and I have started on Vector Spaces. I understand that a Vector space V over the set of real numbers is a set equipped with two operations, namely vector addition and scalar multiplication which follow the rules of algebra.

Now I'm stuck on one definition, specifically the vector subspace which goes as follows:

Let V be a vector space. A vector subspace W is a subset of vector V, which is itself a vector space, with vector addition and scalar multiplication in W being the restrict of those operations in V.

I understand that W is also a vector space and it also follows vector addition and scalar multiplication. The part that I do no understand is this:

"being the restrict of those operations in V"

What does the definition mean by that? I apologize in advance if the question is irrelevant, but I'm sure I'm not getting it because I don't really speak English.

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Here's an example of a vector space $W$, and a subset $V$ of $W$ that is too a vector space.

Let $W = \Bbb R$. With the usual addition and multiplication this is an $\Bbb R$-vector space. Now let $V\subset \Bbb R$ defined as $V = \Bbb R_{>0} := \{x\in\Bbb R : x > 0\}$. Now define operations in $V$ as $a\oplus b:= a\cdot b$ (normal multiplication), and $\lambda\odot a = a^{\lambda}$. Here $a,b \in V$, and $\lambda\in\Bbb R$.

You can verify that $V$ is a subset of $W$ that is also an $\Bbb R$-vector space. However any properties that the operations of $W$ exhibit will have little bearing on what happens in $V$. We don't want this kind of behaviour since sub-structures should "inherit" properties from the larger space (this goes in general for mathematics). So, along with $V$ being a susbet of $W$, and being a vector space, to make it a vector sub-space we require that the operations be the same ones as in $W$, restricted to $V$.

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It simply means that the operations are the same as defined in the space $V$, but the results of this operations, performed on elements of $W$, must be elements of $W$.

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  • $\begingroup$ Aha I understand it. Thanks $\endgroup$ – nTuply May 23 '15 at 21:22
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It just means that you work only in $W$, i.e only add and "scalar-multiply" vectors of $W$. A restriction of a function (remember the vector addition is a function from $V\times V $ into $V$) means you make the domain smaller, that is restrict the function into a subset of the domain.

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