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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
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Alka gupta
this theorem is not applicable for all linear transformation......see an example
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Alka gupta
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yes, if between two vector spaces a LT is defined and dim(V) = dim(V') and we want to check it is an isomorphism or not then in this case it is sufficient to check only one-one no need to check onto
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Deepak singh 1 Best Answer
These are two different things . Theorem is on vector space not Linear transformation . Refer attached explanation
![cropped8636666518035922754.jpg]()
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the thm is saying isomorph. if same dim., what is that?
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clearly , take any two finite dimensional vector space , if their dimension is same then they will be isomorphic .. eg take R^2 and P1(R) {polynomial of degree atmost 1} .
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In Anku's question, we have R3->R3 i.e. same dimension so acc. to the thm. it will be isomorphic, right? and if isomorphic then it implies 1-1, onto, & LT, isn't it?
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check this L.T , it is neither 1-1 nor onto..
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Alka gupta![best-answer]()
this theorem is not applicable for all Linear transformation....see this example