Linear Algebra: Understanding Matrix as a Linear Transformation

[EnglsihLearner]

The matrix is an example of a Linear Transformation, because it takes one vector and turns it into another in a "linear" way.

Hi could you explain it what exactly the bold part suggest?

What is "another in a "linear" way"?

 

[WannabeNewton]

Actually any linear map between two finite dimensional vector spaces (over the same field) can be represented as a matrix so matrices aren't exactly "special" in that sense. A linear map $L: V \rightarrow W$ sends vectors in $V$ to vectors in $W$ such that $L$ preserves addition and scalar multiplication from one vector space to the other.

 

[Fredrik]

A linear transformation is a map $f:U\to V$ such that U and V are vector spaces over the same field F (typically, $F=\mathbb R$ or $F=\mathbb C$), and
$$f(ax+by)=af(x)+bf(y)$$ for all $x,y\in X$ and all $a,b\in F$. If A is an n×n matrix, then the map $x\mapsto Ax$ is linear, because
$$A(ax+by)=aAx+bAy$$ for all n×1 matrices x,y and all real (or complex) numbers a,b.

Note that we can define a function f by defining f(x)=Ax for all n×1 matrices x. Since both the domain and codomain of f is a set whose elements are n×1 matrices to n×1 matrices, it can be said to "take a vector and turn it into another vector". When they say that it does so "in a linear way", they mean that the function is linear in the sense defined above.