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<TITLE> Linear Algebra : Function In Language C. </TITLE>
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<A HREF="http://groups.yahoo.com/group/mathc/">Mathc</A> : The group of this work.
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<A HREF="http://www.geocities.com/xhungab/calculus.html">Calculus</A>,
<A HREF="http://www.geocities.com/xhungab/gnuplot.html">Gnuplot</A>. The other packages,
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<H4><U>Linear Algebra</U> : Language C (version 8).</H4>
The purpose of this work is to verify with numeric </br>
applications, some properties of the linear algebra.</br>
It is a set of functions write in language C.</P>
* You need a C compiler to compile the code (source).</br>
* For this work, I use
<A HREF="http://www.simtel.net/pub/pd/17456.html">Dev-C++ 4</A>,
and
<A HREF="http://www.delorie.com/djgpp/zip-picker.html">DJGPP</A> two freewares. </br>
* The graphic interface is gnuplot.</br>
* You can download this freeware here : <A HREF="http://www.gnuplot.info/">Gnuplot Central</A>. </br>
* My work is also a Freeware.</br>
* Windows, Linux.</br>
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<A HREF="http://www.geocities.com/xhungab/linear/matrxf.zip">matrxf.zip</A>
: You can work with fractions.</P>
addm, subm, multm, powm, smultm, transpose, trace,det, minor, mminor,</br>
cofactor, mcofactor,adjoint,inverse(adjoint), inverse(gaussjordan),</br> inverse(identity matrix), gauss, gaussjordan, LU.</P>
norm, distance, innerproduct,coldim, rowdim, rank, nullity, leastsqrs</P>
<A HREF="http://www.geocities.com/xhungab/linear/matrxg.zip">matrxg.zip</A>
: You can also work with integers.</P>
<A HREF="http://www.geocities.com/xhungab/linear/matrxh.zip">matrxh.zip</A>
: The print, copy, rand, ... functions.</P>
<A HREF="http://www.geocities.com/xhungab/linear/matrxj.zip">matrxj.zip</A>
: System of equations with free variables.</P>
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<H4> In this section, the size of the matrices, are randomly selected</br>
by the computer, but you can selecte the size if you want. </P>
</H4>
<H4><U>Verify with numeric applications</U>:</H4>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxaa.zip">mtrxaa.zip</A>
: How to use the basic functions.</P>
when the size of the matrices are randomly selected by the computer.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxab.zip">mtrxab.zip</A> </P>
The properties of matrix arithmetic.</P>
A+B = B+A </br>
(A+B)+C = A+(B+C)</br>
(AB)C = A(BC)</br>
A(B+C) = AB+AC</br>
(B+C)A = BA+CA</br>
A(B-C) = AB-AC</br>
(B-C)A = BA-CA</br>
a(B+C) = aB+aC</br>
a(B-C) = aB-aC</br>
(a+b)C = aC+bC</br>
(a-b)C = aC-bC</br>
a(bC) = (ab)C</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxac.zip">mtrxac.zip</A> :</P>
* The properties of zero matrices.</br>
* The properties of the transpose.</br>
* The theorem of transpose.</br>
* The theorem of inverse matrices. </P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxad.zip">mtrxad.zip</A> :</P>
* (A+B)**2.</br>
* (A-B)**2.</br>
* (A-B) (A+B).</br>
* Power and inverse.</br>
* Symetric and Skew-Symetric matrices.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxae.zip">mtrxae.zip</A> :</P>
* Solving linear systems by matrix inversions.</br>
* Linear systems with common coefficient matrix.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxaf.zip">mtrxaf.zip</A> :</P>
* The system of equation Ax = b is consistent.</br>
* Inverses of symmetric matrices.</br>
* Power, inverse of diagonal matrices</br>
* Multiply, inverse of triangular matrices.</br>
* Trace property. </P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxag.zip">mtrxag.zip</A></P>
The value of the determinant of </P>
* a diagonal matrix</br>
* a triangular matrix (upper, lower)</br>
* a basic matrix.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxah.zip">mtrxah.zip</A> </P>
LU decomposition :</P>
* LU decomposition.</br>
* Det(A) = Det(L).</br>
* invA = invU invL.</br>
* LU decomposition are not unique.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectaa.zip">vectaa.zip</A> :</P>
Some vector space axioms on (rows,</br>
columns) vectors and on polynomials.</P>
* u + v = v + u </br>
* (u + v) + w = u + (v + w) </br>
* 0 + u = u + 0 = u </br>
* u + (-u) = (-u) + u = 0 </br>
* k( u+ v) = ku + kv </br>
* (k + l) u = ku + lu </br>
* k (lu) = (kl) u </P>
<A HREF="http://www.geocities.com/xhungab/linear/vectab.zip">vectab.zip</A> :</P>
* Properties of Euclidian inner product in R**n.</br>
* Properties of length in R**n.</br>
* Properties of distance in R**n.</br>
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2. </br>
* Cauchy-Schwarz inequality in R**n.</br>
* If u.v =0 : ||u+v||**2 = ||u||**2 + ||v||**2.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectac.zip">vectac.zip</A> :</P>
You can see the result in Gnuplot.</P>
* Reflection about the x-axis.</br>
* Reflection about the y-axis.</br>
* Reflection about the line y = x.</br>
* Orthogonal projection on the x-axis. </br>
* Orthogonal projection on the y-axis.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectad.zip">vectad.zip</A> :</P>
You can see the result in Gnuplot.</P>
* Vector2d (vertical horizontal shift).</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectae.zip">vectae.zip</A> :</P>
You can see the result in Gnuplot.</P>
* Reflection about the xy-plan.</br>
* Reflection about the xz-plan.</br>
* Reflection about the yz-plan.</br>
* Orthogonal projection on the xy-plan. </br>
* Orthogonal projection on the xz-plan. </br>
* Orthogonal projection on the yz-plan. </P>
<A HREF="http://www.geocities.com/xhungab/linear/vectag.zip">vectag.zip</A> :</P>
* Linear combination in R**n, Pn.</br>
* Linear combination of two vectors in R**n, Pn.</br>
* Linear combination of three vectors in R**n, Pn.</br>
* Vectors dependant or independant in R**n, Pn.</br>
* Find the coordinate vector of (w)s in R**n, Pn.</br>
* Find the coordinate vector of w. in R**n, Pn.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectah.zip">vectah.zip</A> :</P>
Inner product, norm, Distance in M22, Mnn, Mnm.</P>
* Properties of Euclidian inner product </br>
* Properties of distance.</br>
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectai.zip">vectai.zip</A> :</P>
Inner product, norm, Distance on R**n generated by A..</P>
* Properties of inner product on R**n generated by A.</br>
* Properties of distance on R**n generated by A.</br>
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectaj.zip">vectaj.zip</A> :</P>
Weighed Euclidean Inner Product, norm, distance on R**n.</P>
* Properties of Weighed Euclidean Inner Product on R**n.</br>
* Properties of distance generated by </br>
the Weighed Euclidean Inner Product on R**n.</br>
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2. </P>
<A HREF="http://www.geocities.com/xhungab/linear/vectak.zip">vectak.zip</A> :</P>
The Gram-Schmidt process.</P>
* with the Euclidean inner product in R**n.</br>
* with the inner product in M2x2.</br>
* with the inner product in M3x2.</br>
* with the inner product in M3x3.</br>
* with the Inner Product on R**n generated by A.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectal.zip">vectal.zip</A> :</P>
Least squares, Orthogonal matrices</P>.
* least squares solution of the linear system Ax = b.</br>
* properties of Orthogonal matrices.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectam.zip">vectam.zip</A> :</P>
Eigenvalue, Eigenvector, Cayley-Hamilton theorem.</P>.
* Some properties of Eigenvalue, Eigenvector.</br>
* The properties of the Cayley-Hamilton theorem.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectan.zip">vectan.zip</A> :</P>
Some linear transformation. Is T a linear transformation?</P>
* T(A) = AX.</br>
* T(A) = trace(A).</br>
* T(x1,x2,x3) = (x1 + x2 + x3).</br>
* Matrix Inner Product on Mnxn.</br>
* Euclidean Inner Product on R**n.</br>
* Inner Product on R**n generated by A.</br>
* Weighed Euclidean Inner Product on R**n.</P>
<A HREF="http://www.geocities.com/xhungab/linear/vectao.zip">vectao.zip</A> :</P>
Similarity</P>
* B is similar to A.</br>
* B is similar to A, also A is similar to B.</br>
* A and invPAP have the same determinant.</br>
* A is invertible if and only, if invPAP is invertible.</br>
* A and invPAP have the same rank.</br>
* A and invPAP have the same nullity.</br>
* A and invPAP have the same trace.</br>
* A and invPAP have the same, characteristic equation.</br>
* The eigenvector of invPAP.</br>
* If B is similar to A, then TrpsB and TrpsA are similar.</P>
<HR><H4><U>Application</U>:</H4>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxid.zip">mtrxid.zip</A>
: Identity matrix application I</P>
* Swap two rows. </br>
* The pivot value. </br>
* Eliminate the coefficient below, above, the pivot. </br>
* Gauss Jordan elimination with the help of the identity matrix.</br>
* Inverse of the matrix with the help of the identity matrix.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxic.zip">mtrxic.zip</A>
: Identity matrix application II</P>
* The work on a column in one step. </br>
* All the values below the pivot in one step. </br>
* Application : Gauss elimination. </br>
* All the values above the pivot in one step. </br>
* Application : Gauss Jordan elimination</br>
* Application : Inverse of the matrix</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxgo.zip">mtrxgo.zip</A>
: Geometric application.</P>
You can verify the result into GnuPlot.</P>
* Find the coefficients of a polynome,</br>
that passes through three, four, five points. </P>
* Find the coefficients a, b, c, d, e of a conic,</br>
ax**2 + by**2 + cx + dy + e = 0 </br>
that passes through four points. </P>
* Find the coefficients a, b, c, d of a circle,</br>
a(x**2 + y**2) + bx + cy + d = 0 </br>
that passes through three points. </P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxch.zip">mtrxch.zip</A>
: Chemistry application.</P>
* Find the coefficients of a chemical equation.</P>
<A HREF="http://www.geocities.com/xhungab/linear/mtrxsy.zip">mtrxsy.zip</A>
: Resolve some nonlinear systems of equations.</P>
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