- Column vector -- a matrix with one column, e.g. ⎝⎛357⎠⎞
- Can be thought of as a point in space
- Represents a single solution, e.g. ⎝⎛357⎠⎞→⎩⎪⎨⎪⎧x1=3x2=5x3=7
- Matrices can be added together:
- Matrices must have the same dimensions
- Add corresponding elements together
(2537)+(5738)=(712615)
- Vectors are a subset of matrices so they can be added too:
(23)+(37)=(510)
- Can multiply every element of a vector by a constant:
5(35)=(1525)
- Negative scalars will also flip the direction of the vector
Span{v1,v2}={c1v1+c2v2 (c1,c2 are scalars)}
- The span of a set of vectors is the set of all possible linear combinations of those vectors.
- Size the of span is infinite unless v1=v2...=vn=0
- If one vector is a linear combination of other vectors in the set, then removing it from the set won't change the span at all.
For all u,v,w in Rn, and c,d in R:
u+v(u+v)+wu+0=0+uu+(−u)c(u+v)(c+d)u(cd)u1u=v+u=u+(v+w)=u=0=cu+cv=cu+du=c(du)=u
ux1+vx2=w has a solution iff w is in Span{u,v}.
Is (13) in the span of (11) and (22)?
(11)x1+(22)x2=(13)(112213)→(102012)
The system has no solution; therefore (13) is not in the span.