- #1
Sasor
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Hey guys, I'm studying these concepts in linear algebra right now and I was wanting to confirm that my interpretation of it was correct.
One to one in algebra means that for every y value, there is only 1 x value for that y value- as in- a function must pass the horizontal line test (Even functions, trig functions would fail (not 1-1), for example, but odd functions would pass (1-1))
Onto means that in a function, every single y value is used, so again, trig and event functions would fail, but odd functions would pass- Any kind of function with a vertical asymptote would pass
So i tried to put these concepts in the context of linear functions and this is what I'm thinking-
Since transformations are represented by matrices,
Linearly independent transformation matrices would be considered one to one- because they have a unique solution. Linearly dependent transformations would not be one-to-one because they have multiple solutions to each y(=b) value, so you could have multiple x values for b
Now for onto, I feel like if a linear transformation spans the codomain it's in, then that means that all b values are used, so it is onto.
Examples:
1-1 but not onto
A linearly independent transformation from R3->R4 that ends up spanning only a plane in R4
Onto but not 1-1
A linearly dependent transformation from R3->R2 that's spans R2
1-1 AND onto
A linearly independent transformation from R3->R3 that spans R3
Neither 1-1 nor onto
A linearly dependent transformation from R2->R2 that spans a line
Is this interpretation correct?
One to one in algebra means that for every y value, there is only 1 x value for that y value- as in- a function must pass the horizontal line test (Even functions, trig functions would fail (not 1-1), for example, but odd functions would pass (1-1))
Onto means that in a function, every single y value is used, so again, trig and event functions would fail, but odd functions would pass- Any kind of function with a vertical asymptote would pass
So i tried to put these concepts in the context of linear functions and this is what I'm thinking-
Since transformations are represented by matrices,
Linearly independent transformation matrices would be considered one to one- because they have a unique solution. Linearly dependent transformations would not be one-to-one because they have multiple solutions to each y(=b) value, so you could have multiple x values for b
Now for onto, I feel like if a linear transformation spans the codomain it's in, then that means that all b values are used, so it is onto.
Examples:
1-1 but not onto
A linearly independent transformation from R3->R4 that ends up spanning only a plane in R4
Onto but not 1-1
A linearly dependent transformation from R3->R2 that's spans R2
1-1 AND onto
A linearly independent transformation from R3->R3 that spans R3
Neither 1-1 nor onto
A linearly dependent transformation from R2->R2 that spans a line
Is this interpretation correct?