My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct.(adsbygoogle = window.adsbygoogle || []).push({});

By definition, a set of vectors S of R^{n}is called a subspace of R^{n}iff for all vectors (I will call them x):

1) (x+y) [tex]\in[/tex] S and

2) kx [tex]\in[/tex] S.

Also, the solution set of a homogeneous system is always a subspace.

When I encounter problems such as determine whether or not {(x_{1},x_{2},x_{3})|x_{1}x_{2}=0} is a subspace of its corresponding R^{n}, I would approach this problem as such:

Since the set has only three vectors, then it's in R^{3}, first of all; then I check for 1): Suppose a set of vectors Y {(y_{1},y_{2},y_{3})|y_{1}y_{2}=0}, then (S+Y)=x_{1}x_{2}+y_{1}y_{2}=0+0=0; for 2): kS=(kx_{1})(kx_{2})=(k0)(k0)=0. Therefore it is a subspace of R^{3}.

Is my way of solving this problem correct?

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# Determine whether or not something is a subspace

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