(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether the following mappings f is onto or one-to-one. Is f an isomorphism?

a) f maps R2 into R2 and is defined by f(x,y) = (x-2y, x+y)

b) f maps R2 into R3 and is defined by f(x,y) = (x, y, x+y)

i) f maps R3 into P2(R), defined by f(a1, a2, a3) = a2 - a3x + (1-a1)x^2

j) f maps Rmxn into Rnxm and is defined by f(A) = A^T (transpose of A) for all A in Rmxn

2. Relevant equations

3. The attempt at a solution

Can I solve this by forming the matrix of the output?

ie I have x-2y = 0 and x+y = 0 as a matrix and row reduce, I get a 2x2 identity matrix and therefore

1) There are always solution for any augmented matrix, so f is onto

2) There are no parameters in the matrix and both variables always have a unique solution for every augmented matrix, so f is one-to-one

Is that right?

For b)

I get form the 3x2 matrix and row reduce, then a have third row of zero and no parameter, therefore f is not onto since there will always be some component of R3 that has no solution (hence the zero row). But f is 1-1 since both variables always have a unique solution for any augmented matrix (hence it has 2 leading ones).

Does that explanation seem right?

For i)

I form the matrix and row reduce to a I3x3, therefore f is onto and 1-1.

For j)

f is one-to-one since it forms the transpose of A and every row of A is mapped to every column of the image once.

It is onto since this applies to every matrix of Rmxn, therefore transposed into every matrix of Rnxm.

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# Homework Help: Linear Algebra - isomorphism questions

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