91Ó°ÊÓ

Question: In Problems 2 to 4, find out whether the given vectors are dependent or independent; if they are dependent, find a linearly independent subset. Write each of the given vectors as a linear combination of the independent vectors.

$\mathbf{\text{2.}}\left(\text{1,}-\text{2,3}\right)\mathbf{\text{,}}\left(\text{1,1,1}\right)\mathbf{\text{,}}\left(-\text{2,1,}-\text{4}\right)\mathbf{\text{,}}\left(\text{3,0,5}\right)$

Let $\mathbf{A}\mathbf{=}\mathbf{2}\hat{\mathbf{i}}\mathbf{-}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\hat{\mathbf{k}}$ . (a) Find a unit vector in the same direction as A . Hint: Divide A by $\left|A\right|$. (b) Find a vector in the same direction as A but of magnitude 12 . (c) Find a vector perpendicular to A . Hint: There are many such vectors; you are to find one of them. (d) Find a unit vector perpendicular to A . See hint in (a).

Find the characteristic frequencies and the characteristic modes of vibration for systems of masses and springs as in Figure 12.1 and Examples 3,4 , and 6 for the following arrays.

5k,m,2k,m,2k

Use Cramer's rule to solve for x and t the Lorentz equations of special relativity:

$\{\begin{array}{l}{x}^{\cancel{c}}=\mathrm{γ}(x-vt)\\ t^{\cancel{c}}=\mathrm{γ}\left(t-vx/c^2\right)\end{array}$

where${\mathit{γ}}^{\mathbf{2}}\left(1-v^2/c^2\right)\mathbf{=}\mathbf{1}$

Caution: Arrange the equations in standard form.

Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

(a) Prove that $\mathbf{Tr}\mathbf{\left(}\mathbf{\text{AB}}\mathbf{\right)}\mathbf{=}\mathbf{Tr}\mathbf{\left(}\mathbf{\text{BA}}\mathbf{\right)}$. Hint: See proof of (9.13).

(b) Construct matrices A, B, Cfor which $\mathbf{Tr}\mathbf{\left(}\mathbf{\text{ABC}}\mathbf{\right)}\mathbf{≠}\mathbf{Tr}\mathbf{\left(}\mathbf{\text{CBA}}\mathbf{\right)}$, but verify that $\mathbf{Tr}\mathbf{\left(}\mathbf{\text{ABC}}\mathbf{\right)}\mathbf{=}\mathbf{Tr}\mathbf{\left(}\mathbf{\text{CAB}}\mathbf{\right)}$.

(c) If Sis a symmetric matrix and Ais an antisymmetric matrix, show that$\mathbf{Tr}\mathbf{\left(}\mathbf{\text{SA}}\mathbf{\right)}\mathbf{=}\mathbf{0}$. Hint: Consider$\mathbf{Tr}\mathbf{(}\mathbf{\text{SA}}{\mathbf{)}}^{\mathbf{\text{T}}}$and prove that$\mathbf{Tr}\mathbf{\left(}\mathbf{\text{SA}}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{Tr}\mathbf{\left(}\mathbf{\text{SA}}\mathbf{\right)}$.

Square $\mathbf{(}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{B}\mathbf{)}$; interpret your result geometrically. Hint: Your answer is a law which you learned in trigonometry.

Let each of the following matrices represent an active transformation of vectors in ( x , y )plane (axes fixed, vector rotated or reflected). As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection.

$\frac{\mathbf{1}}{\mathbf{2}}\left[\begin{array}{cc}-\sqrt{3}& 1\\ -1& -\sqrt{3}\end{array}\right]$

As in Problem 24, find the equations of the line intersections of the planes in Problem 23. Find the distance from the point (1,0,0) to the line.

Show that ifA and Bare matrices which don't commute, then ${\mathbf{e}}^{\left(A+B\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{A}}{\mathbf{e}}^{\mathbf{B}}$ , but if they do commute then the relation holds. Hint: Write out several terms of the infinite series for ${\mathbf{e}}^{\mathbf{A}}{\mathbf{e}}^{\mathbf{B}}$ , and ${\mathbf{e}}^{(A+B)}$and, do the multiplications carefully assuming that anddon't commute. Then see what happens if they do commute