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In the vector space for all real-valued functions, find a basis for the subspace spanned by \(\left\{ {{\bf{sin}}t,\,{\bf{sin2}}t,\,{\bf{sin}}t\,{\bf{cos}}t} \right\}\).

Short Answer

Expert verified

\(\left\{ {\sin t,\sin t\cos t} \right\}\)and \(\left\{ {\sin t,\;\sin 2t} \right\}\)

Step by step solution

01

Find the set of vectors for V

Let \({v_1} = \sin t\), \({v_2} = \sin 2t\), \({v_3} = \sin t\cos t\).

Simplify the equation \({v_3} = \sin t\cos t\) using trigonometric identities.

\[\begin{array}{c}{v_3} = \frac{1}{2}\left( {2\sin t\cos t} \right)\\ = \frac{1}{2}\sin 2t\\ = \frac{1}{2}{v_2}\end{array}\]

So, the vectors \({v_2}\) and \({v_3}\) are dependent.

02

Write the spanning set

The spanning set reduces as shown below:

\(\begin{array}{c}V = {\rm{span}}\left\{ {{v_1},\,{v_2},\,{v_3}} \right\}\\ = {\rm{span}}\left\{ {{v_1},{v_2}} \right\}\;\;{\rm{or}}\\ = {\rm{span}}\left\{ {{v_1},{v_3}} \right\}\end{array}\)

So, the above set represents the basis of H.

So, the basis are \(\left\{ {\sin t,\sin t\cos t} \right\}\) and \(\left\{ {\sin t,\;\sin 2t} \right\}\).

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Most popular questions from this chapter

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.

(M) Show that \(\left\{ {t,sin\,t,cos\,{\bf{2}}t,sin\,t\,cos\,t} \right\}\) is a linearly independent set of functions defined on \(\mathbb{R}\). Start by assuming that

\({c_{\bf{1}}} \cdot t + {c_{\bf{2}}} \cdot sin\,t + {c_{\bf{3}}} \cdot cos\,{\bf{2}}t + {c_{\bf{4}}} \cdot sin\,t\,cos\,t = {\bf{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\bf{5}} \right)\)

Equation (5) must hold for all real t, so choose several specific values of t (say, \(t = {\bf{0}},\,.{\bf{1}},\,.{\bf{2}}\)) until you get a system of enough equations to determine that the \({c_j}\) must be zero.

(M) Determine whether w is in the column space of \(A\), the null space of \(A\), or both, where

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\)

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).

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