Gaussian Elimination
- When hyperplanes intersect, the common points will always be in lower dimension hyperplane until they are reduced.
Balancing a chemical formula
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Given the equation:
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All constant terms are zero in this case and hence this is homogeneous system of equations)
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All homogeneous system of equations have a trivial solution (i.e all variables can be zero)
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We are interested in the non-trivial solution, which exists if there is free variable. This results in infinite number of non-trivial solutions)
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Whenever we have the number of equations less than number of variables, then we are likely to have infinite solutions. (unless there is some inconsistency or contradictions)
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Inconsistency means the LHS and RHS do not match. For example LHS has all zero values and RHS has non-zero value.
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The rank of the matrix is defined to be r. # of unique
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Number of free variables = n - r
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If there is free variable, it can have any values
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We want w, x,y,z should be smallest possible +ve whole number.
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So picking 6 for z would meet this requirement. Thus w = 4,x = 3, y = 2 and the equation is balanced.
System of m linear equations in n variables
- In RREF, assuming no inconsistency detected, if the # of leading entries = r, the rank of the matrix is defined to be r
- Number of free variables = n - r
- if n-r > 0, i.e r < n, system has infinitely many solutions.
- if r = n, system has a unique solution.
- r cannot be greater than n
\[ \left[\hspace{5pt}\begin{array}{cc | c} 1 & 2 & 3 \\\ 4 & 5 & 9 \end{array}\hspace{5pt}\right] \]
\[\begin{bmatrix} a & b \\\ c & d \end{bmatrix} \]
\begin{alignat} 10&x+&3&y=2\\\ 3&x+&13&y=4 \end{alignat}
\[\begin{bmatrix} a_{11} & b_{11} & … & a_n & \vert & c \\\ d & e & … & d_n & \vert & f \end{bmatrix}\]
\[ \begin{equation} {3 H2 + N2O -> 2 NH3 + H2O} \end{equation}\]