Surjectivity and linear maps question

[dyanmcc]

In my head this proof seems obvious, but I am unable to write it rigorously. Any help would be appreciated!

Prove that it T is a linear map from F^4 to F^2 such that

kernel T ={(x1, x2, x3, x4) belonging to F^4 | x1 = 5x2 and x3 = 7x4}, then T is surjective.

 

[matt grime]

What's the dimension of the image? (think in terms of the dimension of the domain and the dimension of the kernel)

 

[dyanmcc]

the dimension of the kernel is two, the dimension of the range is two

so F^2 equals dim range and therefore is surjective?

 

[dyanmcc]

dimension of kernel equal 2. Dimension of range equals 2.

dimension of domain equals 4. Since dim range = 2 and F^2 is the whole space of the range, then it is surjective?

 

[matt grime]

F^2 equals dim range and therefore is surjective?

F^2 is a vector space, so it can't equal a dimension.

The image is a 2-d subspace of a 2-d space, so it is all of it.