= Linear algebra
{wiki}
<Linear algebra> studies <vector spaces> and <linear maps>: operations that preserve <vector addition> and <scalar multiplication>. For a <linear map> $T$, the defining rule is $T(au+bv)=aT(u)+bT(v)$, where $u,v$ are <vectors> and $a,b$ belong to the underlying <field>. The <vectors> might be arrows, lists of measurements, or <polynomials>. What matters is that their <linear combinations> make sense and that the same rule governs their transformation.
A concrete <linear map> is $T(x,y)=(x+y,x-y)$. Its two outputs record a sum and a difference. Since $T(1,0)=(1,1)$ and $T(0,1)=(1,-1)$, we immediately obtain $T(2,1)=2(1,1)+(1,-1)=(3,1)$. Knowing the outputs of the <basis vectors> determines the output of every <vector>; a <matrix> stores exactly that information in its columns.
The attraction of <linear algebra> is that one small collection of ideas answers several different questions. A <system of linear equations> asks which inputs produce a prescribed output. The <image of a linear map> describes the attainable outputs, while the <kernel> describes the invisible input changes. A <basis> gives every <vector> an address, and <change of basis> lets us choose addresses that make a <linear map> easier to understand. This shared language turns a collection of separate calculations into a picture of what is possible and how much freedom remains. Gilbert Strang's MIT course emphasizes these connections and their usefulness across other disciplines.https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/[Strang, MIT 18.06: Linear Algebra]{ref}
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