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I have a simple question I have to answer but I am not sure where to start with this due to my lack of experience regarding subspaces. Can anybody help me?

  • Assume $V \subset \Bbb R^n$ is a linear subspace. Show that $0 ∈ V$.

  • Assume that $V, W \subset \Bbb R^n$ are linear subspaces. Show that $V \cap W$ is a linear subspace of $\Bbb R^n$ as well.

Community
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Dan
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2 Answers2

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I'll help you with the first one. First, we suppose that the only element of $V$ is a zero vector. Everything is ok, we can check the axioms of vector space, $V$ is indeed a vector space. Then, suppose we have $x\in V$, $x\ne 0$, $V$ - vector space. By axioms of vector space, we can can multiply $x$ by any scalar number and the result will still be in $V$; let's say that we chose $-1$; therefore, $-x\in V$. On the other hand, if two vectors are in a vector space, then their sum is in this vector space, too (it's also an axiom of vector space). We take the sum: $x+ (-x) = 0$. We conclude that $0\in V$.

Another way to conclude the same thing is to chose a scalar equal to zero.

TZakrevskiy
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You need to use the definition/criteria of what it means to be a linear subspace, and the definition of the intersection of two sets (here, the intersection of two subspaces).

For the first: One of the criteria of a subspace is that it contains the zero vector. So by definition, if we assume $V$ is a subspace of $\mathbb R^n$, then it must follow that $0 \in V$.

For the second, again, use the definition of linear subspace. We assume $V, W \subset R^n$ are linear subspaces. By definition, $0 \in V$ and $0 \in W$. So what can you conclude about $0 \in V \cap W$? Similarly, show that $V \cap W$ satisfies the criteria of a subspace of $R^n$. Informally, the intersection of two subspaces keeps all and only those vectors that are in both of the subspaces. So if two vectors are in $V\cap W$, then those vectors are in each subspace $V$ and $W$, thus their sum is in each subspace, so their sum is in the intersection, $V\cap W$.

amWhy
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  • That is what I don't understand about the first one since it is by definition that if $V$ is a linear subspace, then $0 \in V$ must be true. So how would I show it? – Dan Oct 31 '13 at 13:17
  • Exactly, you simply appeal to the definition. If we assume $V$ is a subspace of $R^n$, then $V$ must contain the zero vector of $R^n$, hence $0\in V$. – amWhy Oct 31 '13 at 13:18
  • Some books give the condition that $V\ne\emptyset$ rather than $\vec0\in V$ in the definition of a subspace. So in this case there is at least something to prove. – Martin Sleziak Nov 05 '13 at 08:27