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Chapter 4: Q27E (page 191)
In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
\({\bf{1}} + {\bf{2}}{t^{\bf{3}}}\),\({\bf{2}} + t - {\bf{3}}{t^{\bf{2}}}\),\(t + {\bf{2}}{t^2} - {t^{\bf{3}}}\)
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Define a linear transformation by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 0 \right)}\end{array}} \right)\). Find \(T:{{\mathop{\rm P}\nolimits} _2} \to {\mathbb{R}^2}\)polynomials \({{\mathop{\rm p}\nolimits} _1}\) and \({{\mathop{\rm p}\nolimits} _2}\) in \({{\mathop{\rm P}\nolimits} _2}\) that span the kernel of T, and describe the range of T.
Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying
figure. The vectors\(\left( {\begin{array}{*{20}{c}}{2.6}\\{ - 1.5}\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\3\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}0\\0\\{4.8}\end{array}} \right)\)in\({\mathbb{R}^{\bf{3}}}\)form a basis for the unit cell shown on the right. The numbers here are Angstrom units\(\left( {1\mathop { A}\limits^{{\rm{ o}}} = 1{0^{ - 8}}cm} \right)\). In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedralsites (so named because of the geometric objects
formed by atoms at these locations).
The hexagonal close-packed lattice and its unit cell.
37. One of the octahedral sites is\(\left( {\begin{array}{*{20}{c}}{1/2}\\{1/4}\\{1/6}\end{array}} \right)\), relative to the lattice basis. Determine the coordinates of this site relative to the standard basis of\({\mathbb{R}^{\bf{3}}}\).
Justify the following equalities:
a.\({\rm{dim Row }}A{\rm{ + dim Nul }}A = n{\rm{ }}\)
b.\({\rm{dim Col }}A{\rm{ + dim Nul }}{A^T} = m\)
Let \(H\) be an \(n\)-dimensional subspace of an \(n\)-dimensional vector space \(V\). Show that \(H = V\).
Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.
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