# What characterizes a linear transformation?

Consider a linear transformation of a vector $\mathbf{x}$, $T(\mathbf{x})\rightarrow \mathbf{y}$, described by a matrix $\mathbf{A}$ i-e $\mathbf{Ax}=\mathbf{y}$. The matrix $\mathbf{A}$ can be a square $n\times n$ or a $n \times m$ rectangular matrix.

My question is, what are important ingredients of a linear transformation. How can we use properties of $\mathbf{A}$ like rank, column space $C(\mathbf{A})$, etc to explain what $\mathbf{A}$ is doing to $\mathbf{x}$.

• Are u referring to $nxn$ and $nxm$ as matrices of your linear transformation? – Bruno Reis Sep 29 '16 at 15:03
• I am referring to $\mathbf{A}$. $\mathbf{Ax}=\mathbf{y}$ where $\mathbf{x,y}$ are column vectors – NAASI Sep 29 '16 at 15:20

The way I visualize the $A$ matrix of a isomorphic transformation is the following:

We have:

$$A: V\rightarrow W\\ B_v = \left\{ V_1,V_2,...,V_n \right\} \\ B_w = \left\{ W_1,W_2,...,W_n \right\} \\ x \in V\\y \in W$$

With that in hands, we know that: $$x = a_1V_1+a_2V_2+...+a_nV_n\\ A(x) = a_1A(V_1)+a_2A(V_2)+...+a_nA(V_n)\\ A(x) = a_1W_1+a_2W_2+...+a_nWn\\$$

That last line can be written as: $$Ax = y \Rightarrow \begin{pmatrix} \vec{w_1} & \vec{w_2} & \cdots & \vec{w_n} \\ \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots \\ \end{pmatrix} \begin{pmatrix} a_1\\ a_2\\ \vdots\\ a_n \end{pmatrix}$$

And by that we can assume that our columns fo the matrix $A$ are the transformed basis vectors of our domain basis $B_v$.

First thing is to understand linear transfomations without using the crutch of matrices.

Important thing is it is function from one vector space to another. Vector spaces are sets with which are abelian groups, and admitting scalar multiplication satisfying a list of conditions.

A function $T\colon V\to W$ is given between two vector spaces $V,W$ Given two vectors $v,v'\in V$, one can add them using the addition of $V$ and then apply $T$ to their sum getting $T(v+v')$. ----(*)

Or one can apply $T$ to both vectors getting two vectors of $W$, namely $T(v), T(v')$. These two can be added using the addition in $W$ which is the vector $T(v)+T(v')$ -----(**)

If the results in (*) and (**) are equal, and similarly if $T(av) = T(v)$, (scalar multiply and then apply T, or apply T then scalar multiply) we call the function $T$ a linear transformation.

Note that the definition does not use the word like rank, subspaces, determinant etc.

• Thanks for your reply. I am particularly interested in understanding the concept of LT from the perspective of matrices. That is why I asked it in $Ax=y$ form – NAASI Sep 29 '16 at 15:58
• In matrix terms: $Ax=b$ ($x,b$ are vectors) has a solution for $x$ iff $b$ is in the column space of $A$; (consistency of a linear system of equations $Ax=b$), Rank tells how many linearly independent vectors are there in the range of $A$ when regarded as a function. – P Vanchinathan Sep 29 '16 at 16:08