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Determine whether or not something is a subspace 
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#1
Mar709, 03:23 PM

P: 37

My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct.
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? 


#2
Mar709, 05:31 PM

P: 34

You are right in that you must check your the two conditions. But you must do it for arbitrary vectors, you didn't seem to do this correctly. Consider
[tex] (1,0,0), (0,1,0) \in \{(x_{1},x_{2},x_{3})x_{1}x_{2}=0\}. [/tex] If you add them you obtain [tex] (1,1,0) [/tex] which clearly does not have [tex] x_{1}x_{2} = 0 [/tex]. Also the way you present your vectors doesn't seem standard. 


#3
Mar709, 05:35 PM

P: 74

{(x1,x2,x3)x1x2=0} is a subset of [tex]\mathbb{R}^3\[tex] which satisfies the property that x1x2 = 0. but since x1, x2 are in [tex]\mathbb{R}[tex], then either x1=0 or x2=0 or both equal zero. To proof that it is a subspace, let y and s be vectors in {(x1,x2,x3)x1x2=0} such that y = (y1,y2,y3) and s = (s1,s2,s3), then y+s = (y1+s1,y2+s2,y3+s3) check (y1+s1)(y2+s2) = y1y2+y1s2+s1y2+s1s2= y1s2+s1y2 (since y1y2=0 and s1s2=0 clear in general; y1s2+s1y2 is not equal to zero. this implies that it does not satisfy the property x1x2 = 0. Hence the set {(x1,x2,x3)x1x2=0} is not a subspace of [tex]\mathbb{R}^3\[tex]. we don't need to proof the second property 


#4
Mar709, 05:39 PM

P: 74

Determine whether or not something is a subspace
{(x1,x2,x3)x1x2=0} is a subset of R3 which satisfies the property that x1x2 = 0. but since x1, x2 are in R3 then either x1=0 or x2=0 or both equal zero. To proof that it is a subspace, let y and s be vectors in {(x1,x2,x3)x1x2=0} such that y = (y1,y2,y3) and s = (s1,s2,s3), then y+s = (y1+s1,y2+s2,y3+s3) check (y1+s1)(y2+s2) = y1y2+y1s2+s1y2+s1s2= y1s2+s1y2 (since y1y2=0 and s1s2=0 clear in general; y1s2+s1y2 is not equal to zero. this implies that it does not satisfy the property x1x2 = 0. Hence the set {(x1,x2,x3)x1x2=0} is not a subspace of R3. we don't need to proof the second property 


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