Exercises

1. Let V be a complex inner product space and let ${\bf u}, {\bf v} \in V .$ Show that
   $$\left\langle {\bf u} , {\bf v} \right\rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{{\bf j} \theta} \left\| {\bf u} + e^{{\bf j} \theta} {\bf v} \right\|^2 {\text d}\theta .$$
2. For the following problems, use Gauss-Jordan Elimination to put the matrices in RREF. $$(a) \quad A= \begin{bmatrix} 2 & -2 & 4 & -2\\ 2 & 1 & 10 & 7\\ -4 & 4 & -8 & 4\\ 4 & -1 & 14 & 6 \end{bmatrix}, \qquad\quad (b) \quad A= \begin{bmatrix} 1 & 3 & 4 & 14\\ 2 & -3 & 2 & 10\\ 3 & -1 & 1 & 9\\ \end{bmatrix}$$
3. If matrix A represents an augmented matrix system of equations of the form $ax_{1}+bx_{2}+cx_{3}=d,$ find the values of x₁, x₂, and x₃. $$(a) \quad A= \begin{bmatrix} 1 & 2 & 2 & 1\\ 2 & 3 & 6 & 0 \\ 3 & 4 & 7 & 0\\ \end{bmatrix}, \qquad\quad (b) \quad A= \begin{bmatrix} 2 & 4 & -1 & -9\\ -1 & 1 & -1 & -6 \\ -3 & 3 & 4 & 3\\ \end{bmatrix}$$

1. For which two numbers a will elimination fail on matrix $\begin{bmatrix} a&1 \\ a&1 \end{bmatrix} ?$
2. For which hree numbers a will elimination fail to give three pivots? $\begin{bmatrix} a&1&4 \\ a&a&3 \\ a&a&a \end{bmatrix} .$

1. Find the matrix for the transformation. $$\mbox{(a)} \quad S \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \right) = \begin{bmatrix} 1x_{1}+4x_{2}-5x_{3}\\ -3x_{1}+3x_{2}-2x_{3}\\ 1x_{1}+-2x_{2}+6x_{3} \end{bmatrix}; \qquad \mbox{(b)} \quad T \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} 5x_{1}+3x_{2}-4x_{3}\\ x_{2}+2x_{3}\\ \end{bmatrix};$$ $$\mbox{(c)} \quad U \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}-x_{2}+x_{3}\\ 3x_{1}+x_{2}-x_{3}\\ 2x_{1}+2x_{2}+x_{3} \end{bmatrix}$$
2. Is the following transformation linear? Check that both axioms T(A+B)=T(A)+T(B) and T(c*A)=c*T(A) are satisfied. $$\mbox{(a)} \quad S \left( \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} \right) = \begin{bmatrix} -x_{1}+3x_{2}\\ 0\\ \end{bmatrix}; \qquad \mbox{(b)} \quad T \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} 2x_{1}-4\\ x_{2}-2x_{3}\\ \end{bmatrix};$$ $$\mbox{(c)} \quad U \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}-x_{2}\\ 3x_{2}+2x_{3}\\ -x_{3}-4x_{4}\\ \end{bmatrix}; \qquad \mbox{(d)} \quad V \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}+3x_{2}+2x_{3}+1x_{5}\\ -x_{1}-x_{2}-x_{3}+x_{4}+x_{6}\\ 4x_{2}+2x_{3}+4x_{4}+3x_{5}+3x_{6}\\ x_{1}+3x_{2}+2x_{3}-2x_{4}\\ \end{bmatrix} .$$
3. Find the composition of transformations S with T, i.e. S∘T.
   1. $$\quad S:\ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} \,\mapsto \, \begin{bmatrix} x_{1}+3x_{2} \\ 6x_{1}+5x_{2} \\ 3x_{1}+8x_{2} \end{bmatrix} \quad T:\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix}\,\mapsto \, \begin{bmatrix} x_{1}+3x_{2}+7x_{3} \\ -x_{1}-2x_{2}+6x_{3} \\ -4x_{1}+x_{2}-7x_{3} \\ \end{bmatrix}$$
   2. $$\quad S:\ \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} \,\mapsto \, \begin{bmatrix} -2x_{1}+3x_{2} \\ 4x_{2} \\ -3x_{1}-2x_{2} \\ x_{1}-2x_{2} \end{bmatrix}, \quad T :\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix}\,\mapsto \, \begin{bmatrix} x_{2}-2x_{3}+x_{4} \\ 2x_{1}+5x_{2}-2x_{3} \\ -x_{1}+2x_{2}-3x_{3}+2x_{4} \end{bmatrix} .$$

Affine Maps

1. Let P, Q, R, S be points in an affine space A such that P − Q = R − S. Show that F(P) − F(Q) = F(R) − F(S) for any affine transformation F.
2. Determine the matrix representation of the affine transformation S : 𝔸 → 𝔸 if 𝔸 = (A, ℝ²) and S(P) = Q where Q = (αx, βy) for P = (x, y). What type of transformation is this?