Does intersection distributes over linear sum
Multi tool use
$begingroup$
Let $V(Bbb F)$ be a vector space over $Bbb F$.L,M,N are subspaces of $V(Bbb F)$ then which of the following are necessarily true ?
1.$Lcap(M+(Lcap N))=(Lcap M)+(Lcap N)$
2.$Lcap(M+(Lcap N))neq(Lcap M)+(Lcap N)$
3.$Lcap(M+N)=(Lcap M)+(Lcap N)$
4.$Lcap(M+N)neq(Lcap M)+(Lcap N)$
My Thought:
I assumed $Bbb R^3(Bbb R)$ and let L,M,N be the planes passing through origin we know that they are subspaces for the assumed space.
let
L=${(x,y,z)in Bbb R^3(Bbb R)|x+y+9z=0}$
M=${(x,y,z)in Bbb R^3(Bbb R)|x+9y+z=0}$
N=${(x,y,z)in Bbb R^3(Bbb R)|9x+y+z=0}$
I calculated as below (these planes will intersect at three distinct straight lines passing through origin.)
Thus
$Lcap M $= space generated by $(-80,8,8)$
$Lcap N $= space generated by $(8,-80,8)$
$ Ncap M $= space generated by $(8,8,-80)$
Let for first option we can see $Lcap N $=$lt(8,-80,8)gt$ and the line $Lcap N $ intesect M at only $(0,0,0)$ thus M+$Lcap N $ is actually $Bbb R^3$ hence $LcapBbb R^3$ is $Bbb R^3$ again on the right hand side $Lcap M $ and $Lcap N $ are two independent straight lines $lt(8,8,-80)gt$ and $lt(8,-80,8)gt$ respectively intersect at origin so their linear sum will be a 2-d space which will not be equal to $Bbb R^3$ .
Hence 2 is correct.
For 3 and 4 I have found posts regarding it.
While checking answer in my book it showed 1 is correct.Please help me
I am also making conclusions based on my intuition please say whether they are wrong or right in general :
1.$Lcap(M+(Lcap N))supset(Lcap M)+(Lcap N)$
2.$Lcap(M+N)supset(Lcap M)+(Lcap N)$
linear-algebra vector-spaces
asked Jan 10 at 14:42
onlymathonlymath
749
$endgroup$
add a comment |
$begingroup$
Let $V(Bbb F)$ be a vector space over $Bbb F$.L,M,N are subspaces of $V(Bbb F)$ then which of the following are necessarily true ?
1.$Lcap(M+(Lcap N))=(Lcap M)+(Lcap N)$
2.$Lcap(M+(Lcap N))neq(Lcap M)+(Lcap N)$
3.$Lcap(M+N)=(Lcap M)+(Lcap N)$
4.$Lcap(M+N)neq(Lcap M)+(Lcap N)$
My Thought:
I assumed $Bbb R^3(Bbb R)$ and let L,M,N be the planes passing through origin we know that they are subspaces for the assumed space.
let
L=${(x,y,z)in Bbb R^3(Bbb R)|x+y+9z=0}$
M=${(x,y,z)in Bbb R^3(Bbb R)|x+9y+z=0}$
N=${(x,y,z)in Bbb R^3(Bbb R)|9x+y+z=0}$
I calculated as below (these planes will intersect at three distinct straight lines passing through origin.)
Thus
$Lcap M $= space generated by $(-80,8,8)$
$Lcap N $= space generated by $(8,-80,8)$
$ Ncap M $= space generated by $(8,8,-80)$
Let for first option we can see $Lcap N $=$lt(8,-80,8)gt$ and the line $Lcap N $ intesect M at only $(0,0,0)$ thus M+$Lcap N $ is actually $Bbb R^3$ hence $LcapBbb R^3$ is $Bbb R^3$ again on the right hand side $Lcap M $ and $Lcap N $ are two independent straight lines $lt(8,8,-80)gt$ and $lt(8,-80,8)gt$ respectively intersect at origin so their linear sum will be a 2-d space which will not be equal to $Bbb R^3$ .
Hence 2 is correct.
For 3 and 4 I have found posts regarding it.
While checking answer in my book it showed 1 is correct.Please help me
I am also making conclusions based on my intuition please say whether they are wrong or right in general :
1.$Lcap(M+(Lcap N))supset(Lcap M)+(Lcap N)$
2.$Lcap(M+N)supset(Lcap M)+(Lcap N)$
linear-algebra vector-spaces
asked Jan 10 at 14:42
onlymathonlymath
749
$endgroup$
add a comment |
$begingroup$
Let $V(Bbb F)$ be a vector space over $Bbb F$.L,M,N are subspaces of $V(Bbb F)$ then which of the following are necessarily true ?
1.$Lcap(M+(Lcap N))=(Lcap M)+(Lcap N)$
2.$Lcap(M+(Lcap N))neq(Lcap M)+(Lcap N)$
3.$Lcap(M+N)=(Lcap M)+(Lcap N)$
4.$Lcap(M+N)neq(Lcap M)+(Lcap N)$
My Thought:
I assumed $Bbb R^3(Bbb R)$ and let L,M,N be the planes passing through origin we know that they are subspaces for the assumed space.
let
L=${(x,y,z)in Bbb R^3(Bbb R)|x+y+9z=0}$
M=${(x,y,z)in Bbb R^3(Bbb R)|x+9y+z=0}$
N=${(x,y,z)in Bbb R^3(Bbb R)|9x+y+z=0}$
I calculated as below (these planes will intersect at three distinct straight lines passing through origin.)
Thus
$Lcap M $= space generated by $(-80,8,8)$
$Lcap N $= space generated by $(8,-80,8)$
$ Ncap M $= space generated by $(8,8,-80)$
Let for first option we can see $Lcap N $=$lt(8,-80,8)gt$ and the line $Lcap N $ intesect M at only $(0,0,0)$ thus M+$Lcap N $ is actually $Bbb R^3$ hence $LcapBbb R^3$ is $Bbb R^3$ again on the right hand side $Lcap M $ and $Lcap N $ are two independent straight lines $lt(8,8,-80)gt$ and $lt(8,-80,8)gt$ respectively intersect at origin so their linear sum will be a 2-d space which will not be equal to $Bbb R^3$ .
Hence 2 is correct.
For 3 and 4 I have found posts regarding it.
While checking answer in my book it showed 1 is correct.Please help me
I am also making conclusions based on my intuition please say whether they are wrong or right in general :
1.$Lcap(M+(Lcap N))supset(Lcap M)+(Lcap N)$
2.$Lcap(M+N)supset(Lcap M)+(Lcap N)$
linear-algebra vector-spaces
asked Jan 10 at 14:42
onlymathonlymath
749
$endgroup$
Let $V(Bbb F)$ be a vector space over $Bbb F$.L,M,N are subspaces of $V(Bbb F)$ then which of the following are necessarily true ?
1.$Lcap(M+(Lcap N))=(Lcap M)+(Lcap N)$
2.$Lcap(M+(Lcap N))neq(Lcap M)+(Lcap N)$
3.$Lcap(M+N)=(Lcap M)+(Lcap N)$
4.$Lcap(M+N)neq(Lcap M)+(Lcap N)$
My Thought:
I assumed $Bbb R^3(Bbb R)$ and let L,M,N be the planes passing through origin we know that they are subspaces for the assumed space.
let
L=${(x,y,z)in Bbb R^3(Bbb R)|x+y+9z=0}$
M=${(x,y,z)in Bbb R^3(Bbb R)|x+9y+z=0}$
N=${(x,y,z)in Bbb R^3(Bbb R)|9x+y+z=0}$
I calculated as below (these planes will intersect at three distinct straight lines passing through origin.)
Thus
$Lcap M $= space generated by $(-80,8,8)$
$Lcap N $= space generated by $(8,-80,8)$
$ Ncap M $= space generated by $(8,8,-80)$
Let for first option we can see $Lcap N $=$lt(8,-80,8)gt$ and the line $Lcap N $ intesect M at only $(0,0,0)$ thus M+$Lcap N $ is actually $Bbb R^3$ hence $LcapBbb R^3$ is $Bbb R^3$ again on the right hand side $Lcap M $ and $Lcap N $ are two independent straight lines $lt(8,8,-80)gt$ and $lt(8,-80,8)gt$ respectively intersect at origin so their linear sum will be a 2-d space which will not be equal to $Bbb R^3$ .
Hence 2 is correct.
For 3 and 4 I have found posts regarding it.
While checking answer in my book it showed 1 is correct.Please help me
I am also making conclusions based on my intuition please say whether they are wrong or right in general :
1.$Lcap(M+(Lcap N))supset(Lcap M)+(Lcap N)$
2.$Lcap(M+N)supset(Lcap M)+(Lcap N)$
linear-algebra vector-spaces
linear-algebra vector-spaces
asked Jan 10 at 14:42
onlymathonlymath
749
asked Jan 10 at 14:42
onlymathonlymath
749
asked Jan 10 at 14:42
onlymathonlymath
749
asked Jan 10 at 14:42
onlymathonlymath
749
asked Jan 10 at 14:42
onlymathonlymath
749
749
add a comment |
add a comment |
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