Consider the plane R 2 and consider any two lines that pass through the origin. Are these lines subspaces of the plane? Are they isomorphic? Are they \"the same\"? Clearly the \"look different\", but mathematically, do they behave the same structurally? Thank you, Please show work Solution 1 . lines cannot be subspaces , they can be made boundaries of a sub space,In rough terms we can call them a subspaces of zero area. 2 . for two graphs to be isomorphic,the relation between output of one graph and input of the other graph must be a bijection.in easy terms for every value of input taken for 2nd graph from the output values of the first, there should be a unique output.as it is a real plane and both are straight lines passiing through orign,both lines are isomorphic 3 . They are absolutely not same but the behaviour will same as i)they have no intercepts ii) they have constant slopes these two propertes make them structurally similar .

1. Consider the plane R 2
and consider any two lines that pass through the origin. Are these lines
subspaces of the plane? Are they isomorphic? Are they "the same"? Clearly the "look
different", but mathematically, do they behave the same structurally?
Thank you, Please show work
Solution
1 . lines cannot be subspaces , they can be made boundaries of a sub space,In rough terms we can
call them a subspaces of zero area. 2 . for two graphs to be isomorphic,the relation between
output of one graph and input of the other graph must be a bijection.in easy terms for every value
of input taken for 2nd graph from the output values of the first, there should be a unique
output.as it is a real plane and both are straight lines passiing through orign,both lines are
isomorphic 3 . They are absolutely not same but the behaviour will same as i)they have no
intercepts ii) they have constant slopes these two propertes make them structurally similar