From our book:

"Suppose that L:

**R^m**-->

**R^n**is a linear transformation. The vector space

**R^m**is called the domain and the vector space

**R^n**is called the range of L. This terminology is just the standard terminology that one uses when working with functions. Notice that if

**v**is a vector in the range of

*f*that there is no guarantee that there is a vector

**u**in the domain of

*f*for which

*f*(

**u**)=

**v**. For example if L:(x,y) --> (x+y+x, 2x, 0) is a linear transformation from

**R^2**to

**R^3**, then there is no vector

**u**in

**R^2**for which

*f*(

**u**)=(0,0,1). The phenomenon also occurs when studying functions of a single variable; for example if

*f*(x)=x^2, then there is no x=a for which

*f*(a)=-2. The definition below makes a distinction between a set that merely contains all possible outputs of a function and the set that actually equals all of these outputs."

What? It specifically states that "

**v**is a vector in the range of

*f*", so how can we even consider that for

*f*(x)=x^2, then there is no x=a for which

*f*(a)=-2? -2 isn't in the range of

*f*...

The book defines

*image*(L) as being a single set that's contained within

*range*(L), but I have no idea how to calculate it.

Can someone give me some pointers?

## Error