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Chapter 4: Q4.5-31E (page 191)

Exercises 31 and 32 concern finite-dimensional vector spaces V and W and a linear transformation \(T:V \to W\).

Let H be a nonzero subspace of V, and let \(T\left( H \right)\) be the set of images of vectors in H. Then \(T\left( H \right)\) is a subspace of W, by Exercise 35 in section 4.2. Prove that \({\bf{dim}}T\left( H \right) \le {\bf{dim}}\left( H \right)\).

Short Answer

Expert verified

\(\dim T\left( H \right) \le p = \dim H\)

Step by step solution

01

Write the transformation vector in subspace H

Let the set \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},....{{\bf{v}}_p}} \right\}\) be a basis for H,i.e. \(\dim H = p\). For any \({\bf{v}}\) in the subspace H:

\(T\left( {\bf{v}} \right) = T\left( {{c_1}{{\bf{v}}_1} + .... + {c_p}{{\bf{v}}_p}} \right)\)

02

Check for statement (b)

Any vector \(T\left( {\bf{v}} \right) \in T\left( H \right)\) is a linear combination of \(T\left( {{{\bf{v}}_1}} \right)\),鈥�.,\(T\left( {{{\bf{v}}_p}} \right)\), i.e.

\(T\left( H \right) = {\rm{span}}\left\{ {T\left( {{{\bf{v}}_1}} \right),.....,T\left( {{{\bf{v}}_p}} \right)} \right\}\)

As \(T\left( {{{\bf{v}}_1}} \right)\),鈥�.,\(T\left( {{{\bf{v}}_p}} \right)\) are not linearly independent, \(\dim T\left( H \right) \le p = \dim H\).

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Most popular questions from this chapter

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 \(V\).

Prove theorem 3 as follows: Given an \(m \times n\) matrix A, an element in \({\mathop{\rm Col}\nolimits} A\) has the form \(Ax\) for some x in \({\mathbb{R}^n}\). Let \(Ax\) and \(A{\mathop{\rm w}\nolimits} \) represent any two vectors in \({\mathop{\rm Col}\nolimits} A\).

  1. Explain why the zero vector is in \({\mathop{\rm Col}\nolimits} A\).
  2. Show that the vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\).
  3. Given a scalar \(c\), show that \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\).

Explain what is wrong with the following discussion: Let \({\bf{f}}\left( t \right) = {\bf{3}} + t\) and \({\bf{g}}\left( t \right) = {\bf{3}}t + {t^{\bf{2}}}\), and note that \({\bf{g}}\left( t \right) = t{\bf{f}}\left( t \right)\). Then, \(\left\{ {{\bf{f}},{\bf{g}}} \right\}\) is linearly dependent because g is a multiple of f.

Justify the following equalities:

a.\({\rm{dim Row }}A{\rm{ + dim Nul }}A = n{\rm{ }}\)

b.\({\rm{dim Col }}A{\rm{ + dim Nul }}{A^T} = m\)

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

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