## ChapterLTLinear Transformations

In the next linear algebra course you take, the first lecture might be a reminder about what a vector space is (DefinitionÂ VS), their ten properties, basic theorems and then some examples. The second lecture would likely be all about linear transformations. While it may seem we have waited a long time to present what must be a central topic, in truth we have already been working with linear transformations for some time.

Functions are important objects in the study of calculus, but have been absent from this course until now (well, not really, it just seems that way). In your study of more advanced mathematics it is nearly impossible to escape the use of functions â€” they are as fundamental as sets are.

## Annotated Acronyms LT.

### TheoremÂ MBLT.

You give me an $$m\times n$$ matrix and I will give you a linear transformation $$\ltdefn{T}{\complex{n}}{\complex{m}}\text{.}$$ This is our first hint that there is some relationship between linear transformations and matrices.

### TheoremÂ MLTCV.

You give me a linear transformation $$\ltdefn{T}{\complex{n}}{\complex{m}}$$ and I will give you an $$m\times n$$ matrix. This is our second hint that there is some relationship between linear transformations and matrices. Generalizing this relationship to arbitrary vector spaces (i.e. not just $$\complex{n}$$ and $$\complex{m}$$) will be the most important idea of ChapterÂ R.

### TheoremÂ LTLC.

A simple idea, and as described in ExerciseÂ LT.T20, equivalent to the DefinitionÂ LT. The statement is really just for convenience, as we will quote this one often.

### TheoremÂ LTDB.

Another simple idea, but a powerful one. â€śIt is enough to know what a linear transformation does to a basis.â€ť At the outset of ChapterÂ R, TheoremÂ VRRB will help us define a very important function, and then TheoremÂ LTDB will allow us to understand that this function is also a linear transformation.

### TheoremÂ KPI.

The pre-image will be an important construction in this chapter, and this is one of the most important descriptions of the pre-image. It should remind you very much of TheoremÂ PSPHS. Also see TheoremÂ RPI, which has a description below.

### TheoremÂ KILT.

Kernels and injective linear transformations are intimately related. This result is the connection. Compare with TheoremÂ RSLT below.

### TheoremÂ ILTB.

Injective linear transformations and linear independence are intimately related. This result is the connection. Compare with TheoremÂ SLTB below.

### TheoremÂ RSLT.

Ranges and surjective linear transformations are intimately related. This result is the connection. Compare with TheoremÂ KILT above.

### TheoremÂ SSRLT.

This theorem provides the most direct way of forming the range of a linear transformation. The resulting spanning set might well be linearly dependent, and beg for some clean-up, but that does not stop us from having very quickly formed a reasonable description of the range. If you find the determination of spanning sets or ranges difficult, this is one worth remembering. You can view this as the analogue of forming a column space by a direct application of DefinitionÂ CSM.

### TheoremÂ SLTB.

Surjective linear transformations and spanning sets are intimately related. This result is the connection. Compare with TheoremÂ ILTB above.

### TheoremÂ RPI.

This is the analogue of TheoremÂ KPI. Membership in the range is equivalent to nonempty pre-images.

### TheoremÂ ILTIS.

Injectivity and surjectivity are independent concepts. You can have one without the other. But when you have both, you get invertibility, a linear transformation that can be run â€śbackwards.â€ť This result might explain the entire structure of the four sections in this chapter.

### TheoremÂ RPNDD.

This is the promised generalization of TheoremÂ RPNC about matrices. So the number of columns of a matrix is the analogue of the dimension of the domain. This will become even more precise in ChapterÂ R. For now, this can be a powerful result for determining dimensions of kernels and ranges, and consequently, the injectivity or surjectivity of linear transformations. Never underestimate a theorem that counts something.