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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

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Given vectors, \({u_{\bf{1}}}\),….,\({u_p}\) and w in V, show that w is a linear combination of \({u_{\bf{1}}}\),….,\({u_p}\) if and only if \({\left( w \right)_B}\) is a linear combination of vectors \({\left( {{{\bf{u}}_{\bf{1}}}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

14. Show that if \(Q\) is an invertible, then \({\mathop{\rm rank}\nolimits} AQ = {\mathop{\rm rank}\nolimits} A\). (Hint: Use Exercise 13 to study \({\mathop{\rm rank}\nolimits} {\left( {AQ} \right)^T}\).)

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)
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