91Ó°ÊÓ

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \left\\{\begin{array}{r} -x+y-z=4 \\ x-y+2 z=3 \\ 2 x-2 y+4 z=6 \end{array}\right. $$

Use vectors to prove the following theorems from geometry; In a parallelogram, the two lines from one corner to the midpoints of the two opposite sides trisect the diagonal they cross,

Write, in parametric form, the equation of the \(y\) axis.

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \begin{array}{r} \mid x-2 y+3 z=0 \\ x+4 y-6 z=0 \\ 2 x+2 y-3 z=0 \end{array} $$

Give numerical examples of : a symmetric matrix; a skew-symmetric matrix; a real matrix; a pure imaginary matrix.

The Pauli spin matrices in quantum mechanics are $$ A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) . \quad B=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad C=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) $$ Show that \(A^{2}=B^{2}=C^{2}=1\). (Note carefully that this 1 means the 2 by 2 unit matrix and not the number 1 ; this notation is customary in quantum mechanics.) Also show that any pair of these matrices anticommute, that is, \(A B=-B A\), etc. Show that the commutator of \(A\) and \(B\), that is, \(A B-B A\), is \(2 i C\), and similarly for other pairs in cyclic order.

Are the following operators linear? Definite integral with respect to \(x\) from 0 to \(1 ;\) the objects being operated on are functions of \(x\).

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \left\\{\begin{aligned} 2 x-y+3 z &=1 \\ 4 x-2 y-z &=-3 \\ 2 x-y-4 z &=-4 \\ 10 x-5 y-6 z &=-10 \end{aligned}\right. $$

Solve the given set of equations by reducing the matrix to echelon form. Sa! geometrically what the solution is (one point, all points on a line or on a plane, or no solution). If the solution is a line, write its vector equation. \(\left\\{\begin{array}{l}5 x+7 y-4 z+w=2 \\ 2 x-y-2 z+3 w=3 \\ 4 x+3 y-5 z+2 w=5 \\ 3 x+2 y+z+5 w=1\end{array}\right.\)

The median of a trapezoid means the line joining the midpoints of the two nonparallel sides. Prove that the median bisects both diagonals; that the median is parallel to the two parallel bases and equal to half the sum of their lengths.