Onto

2. Onto and always existence

Since the equation T(x) = b has solutions ⇔ brangeT, the equation T(x) = b has solutions for any b is equivalent to the following definition.

A transformation T: XY is onto (surjective) if its range is the whole target set Y

Intuitively, we may think of a transformation as a way of "shooting" from source to target. The transformation is onto if any element of the target set is "hit" by some element of the source.

More discussion on onto transformations can be found here.

Example As explained in an earlier example, the Capital City and Age transformations are definitely not onto. For the Instructor transformation to be onto, we need every professor to teach some courses. This is unlikely to happen, so that the Instructor transformation is unlikely to be onto.

From the discussion in this example, the Square transformation is not onto because the range does not include negative numbers. If we change the transformation to Square': R → [0, ∞) (due to different target set, we consider this transformation to be different from Square: RR), then Square' is onto.

Since the range of Reflection in x-axis is the whole target set R2, the transformation is onto. Similarly, the Rotation transformation is also onto.

We also see that

 T[ x1 ] = [ 3x1 + x2 - x3 ] : R3 → R3 x2 x1 - x2 + x3 x3 2x1 + 2x2 + x3

is onto, while

 T[ x1 ] = [ 3x1 + x2 - x3 ] : R3 → R3 x2 x1 - x2 + x3 x3 2x1 - x2 + x3

is not onto.

If a linear transformation T: RnRm is given by a matrix A, then T is onto ⇔ Ax = b has solutions for any right side b. By this criterion, this is equivalent to all rows of A are pivot.

Example To determine whether the linear transformation

 T[ x1 ] x2 x3 x4 x5 x6
=
 [ x1 + 3x2 + 2x3 + x5 ] - x1 - x2 - x3 + x4 + x6 4x2 + 2x3 + 4x4 + 3x5 + 3x6 x1 + 3x2 + 2x3 - 2x4
: R6R4

is onto or not, we find the row echelon form of the corresponding matrix. In this example, the matrix corresponding to T

 [ 1 3 2 0 1 0 ] -1 -1 -1 1 0 1 0 4 2 4 3 3 1 3 2 -2 0 0

has been simplified, by row operations, to

 [ 1 3 2 0 1 0 ] 0 2 1 -1 0 1 0 0 0 2 1 0 0 0 0 0 0 1

Since all rows are pivot, the transformation T is onto.

To determine whether the linear transformation

 S[ x1 ] x2 x3
=
 [ x1 + 3x2 + 2x3 ] - x1 - x2 - x3 4x2 + 2x3 x1 + 3x2 + 2x3
: R3R4

is onto or not, we could certainly appeal to the same computation. However, we do not even need any computation. Since the corresponding 4 equations in 3 variables cannot always have solutions, the transformation is not onto.

We expand on the remark made at the end of the last example and get

A linear transformation T: RnRm is onto ⇒ mn