Serlo: EN: Linear systems and matrices

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A practical example

Linear dependencies often occur in practice. Mathe für Nicht-Freaks: Vorlage:Beispiel

Linear systems and linear maps

As the name suggests, there is an important connection between linear systems of equations and linear maps. For example, let's look at the following linear system

We can combine these two equations into one equation by using the vector notation. The left-hand side is a vector of variables $v,w$, the right-hand side is a constant vector. We can thus write the columns of the system of equations as vectors and obtain

The left-hand side depends on the variables $v$ and $w$. These can be described by a function

This gives us the equation

In particular, $f$ is a linear map.

We have rewritten the above system of equations into an equation for a linear map that results in a constant vector. In other words, the search for a solution to the above system of equations is the same as the search for an image of the right-hand side under $f$. In fact, we obtain the coefficient matrix of the above system of equations as the representation matrix of $f$ with respect to the basis $(1,0{)}^T,(0,1{)}^T$.

More generally, for a linear system with $m$ equations and $n$ variables over a field $K$, we find a corresponding linear map $f:K^n\to K^m$ as follows: Let $A\in K^{m\times n}$ be the coefficient matrix and $b\in K^m$ the right-hand side of the linear system. Then we set $f(x):=Ax$. This function is linear and agrees with the above construction. Then the search for a solution to the system of equations corresponds to the search for a preimage of $b$ under $f$, that is, an $x$ satisfying $f(x)=b$.

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