Many different types of problems can be represented as systems of linear equations, and many operations can be represented as linear transformations. By working in the abstract, we only have to learn a small number of techniques to work with this wide variety of equivalent problems.
Ex. Every quadratic equation can be represented as c0+c1x+c2x2. The only thing that changes between them is the values of the coefficients, which can be represented as a vector ⎝⎛c1c2c3⎠⎞. This implies that we can use linear algebra to answer questions about quadratic equations.
This idea can also be extended to polynomials of any degree. When working with cubics like c0+c1x+c2x2+c3x3, they can be represented by the vector ⎝⎜⎜⎛c0c1c2c3⎠⎟⎟⎞. If you wanted to represent only the quadratics within this vector space, you could use Span{⎝⎜⎜⎛1000⎠⎟⎟⎞,⎝⎜⎜⎛0100⎠⎟⎟⎞,⎝⎜⎜⎛0010⎠⎟⎟⎞}.
This set will only contain points where c3=0, i.e. cubics where the coefficient of x3 is 0, i.e. quadratics. This demonstrates how looking at span can be useful in a concrete sense.
The question of whether Ax=b has a solution is equivalent to a question regarding the span of the columns that make up A.
Let A be an m×n matrix. The following are equivalent:
- For each b in Rn, Ax=b has a solution.
- Each b in Rn is a linear combination of the columns of A.
- The columns of A span Rn.
- A has a pivot position in every row.
A homogeneous system of linear equations has the form Ax=0. They are always consistent (always have at least one solution), since the trivial solution (where every variable equals 0) will always be a solution.
Ex.
⎩⎪⎨⎪⎧3x1+5x2−4x3=0−3x1−2x2+4x3=06x1+x2−8x3=0
→→⎝⎛3−365−21−44−8000⎠⎞⎝⎛30053−9−400000⎠⎞⎝⎛300530−400000⎠⎞
3x2=0→x23x1+5(0)−4x3=0→x1=0=34x3
=⎝⎛34x30x3⎠⎞=x3⎝⎛3401⎠⎞=Span{⎝⎛3401⎠⎞}.