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Chapter 4: Q31E (page 191)

Exercise 31 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let \(T:V \to W\) be a linear transformation, and let \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) be a subset of V.

31. Show that if \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) is linearly dependent in V, then the set of images, \(\left\{ {T\left( {{v_{\bf{1}}}} \right),...,T\left( {{v_p}} \right)} \right\}\), is linearly dependent in W. This fact shows that if a linear transformation maps a set \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) onto a linearly independent set \(\left\{ {T\left( {{v_{\bf{1}}}} \right),...,T\left( {{v_p}} \right)} \right\}\), then the original set is linearly independent, too (because it cannot be linearly dependent).

Short Answer

Expert verified

The set \(\left\{ {T\left( {{v_1}} \right),...,T\left( {{v_p}} \right)} \right\}\) is linearly dependent.

Step by step solution

01

Write the given statement

Suppose \(\left\{ {{v_1},...,{v_p}} \right\}\) is linearly dependent.

02

Use the definition of linear dependence

There exist scalars \({c_1},...,{c_p}\) that are not all zeros, such that

\({c_1}{v_1} + ... + {c_p}{v_p} = 0\).

Take T on both sides.

\(\begin{array}{c}T\left( {{c_1}{v_1} + ... + {c_p}{v_p}} \right) = T\left( 0 \right)\\{c_1}T\left( {{v_1}} \right) + ... + {c_p}T\left( {{v_p}} \right) = 0\end{array}\)

(Since not all \({c_i}\)s are zeros)

03

Draw a conclusion

Thus, \(\left\{ {T\left( {{v_1}} \right),...,T\left( {{v_p}} \right)} \right\}\) is linearly dependent.

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Most popular questions from this chapter

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

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

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.)

Question 11: Let \(S\) be a finite minimal spanning set of a vector space \(V\). That is, \(S\) has the property that if a vector is removed from \(S\), then the new set will no longer span \(V\). Prove that \(S\) must be a basis for \(V\).

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

13. Show that if \(P\) is an invertible \(m \times m\) matrix, then rank\(PA\)=rank\(A\).(Hint: Apply Exercise12 to \(PA\) and \({P^{ - 1}}\left( {PA} \right)\).)
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