Mathc : The news group of this work.
In this section, the size of the vectors,
are randomly selected by the computer,
but you can selecte the size if you want.
Verify with numeric applications:
vectaa01.zip
Some vector space axioms on (rows,
columns) vectors and on polynomials.
* u + v = v + u
* (u + v) + w = u + (v + w)
* 0 + u = u + 0 = u
* u + (-u) = (-u) + u = 0
* k( u+ v) = ku + kv
* (k + l) u = ku + lu
* k (lu) = (kl) u
vectab01.zip :
* Properties of Euclidian inner product in R**n.
* Properties of length in R**n.
* Properties of distance in R**n.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.
* Cauchy-Schwarz inequality in R**n.
* If u.v =0 : ||u+v||**2 = ||u||**2 + ||v||**2.
vectac01.zip :
You can see the result in Gnuplot.
* Reflection about the x-axis.
* Reflection about the y-axis.
* Reflection about the line y = x.
* Orthogonal projection on the x-axis.
* Orthogonal projection on the y-axis.
vectae01.zip :
You can see the result in Gnuplot.
* Reflection about the xy-plan.
* Reflection about the xz-plan.
* Reflection about the yz-plan.
* Orthogonal projection on the xy-plan.
* Orthogonal projection on the xz-plan.
* Orthogonal projection on the yz-plan.
vectag01.zip :
* Linear combination in R**n, Pn.
* Linear combination of two vectors in R**n, Pn.
* Linear combination of three vectors in R**n, Pn.
* Vectors dependant or independant in R**n, Pn.
* Find the coordinate vector of (w)s in R**n, Pn.
* Find the coordinate vector of w. in R**n, Pn.
vectah01.zip :
Inner product, norm, Distance in M22, Mnn, Mnm.
* Properties of Euclidian inner product
* Properties of distance.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.