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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

Suppose a \({\bf{4}} \times {\bf{7}}\) matrix A has four pivot columns. Is \({\bf{Col}}\,A = {\mathbb{R}^{\bf{4}}}\)? Is \({\bf{Nul}}\,A = {\mathbb{R}^{\bf{3}}}\)? Explain your answers.

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{3}}}\)in the plane\(x + {\bf{2}}y + z = {\bf{0}}\). (Hint:Think of the equation as a “system” of homogeneous equations.)

Let be a basis of\({\mathbb{R}^n}\). .Produce a description of an \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)matrix A that implements the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\). Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)). (See Exercise 21.)

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

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