If a linear transformation T: V -> W is surjective, what is the relationship between the dimension of the image and the dimension of W?
Show answer
They are equal
By definition, a transformation is surjective (onto) if its image is the entire codomain W. This implies that the rank of the transformation must equal the dimension of W.
Also asked as:
- In linear algebra, how does the dimension of the image compare to the dimension of the codomain W for a surjective transformation T: V -> W?