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Q29E

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Linear Algebra and its Applications
Found in: Page 191
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Use Exercise 28 to explain why the equation \(Ax = b\)has a solution for all \({\rm{b}}\) in \({\mathbb{R}^m}\) if and only if the equation \({A^T}x = 0\)has only the trivial solution.

For the condition given, the number of non-pivot columns in \({A^T}\) is 0, which implies that \({A^T}x = 0\) has a trivial solution.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Assume an arbitrary system and relate the given statement with it

Consider the nonhomogeneous system \(Ax = b\) with matrix \(A\) of the size \(m \times n\). If the system has a pivot position in each row, it will have solution for all values of \(b\) in the subspace of \({\mathbb{R}^m}\) .

Step 2: Use the result of Exercise 28b

If the system \(Ax = b\) has a pivot position in each row, then the number of columns in \(A\) is \(m\), that is, \({\rm{dimCol}}\,\,A = m\). By the result of Exercise 28b,\({\rm{dimCol}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = m\). Put the values in \({\rm{dimCol}}\,\,A = m\) to get

\(\begin{aligned} {\rm{dim}}\,{\rm{Col}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= m\\m + {\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= m\\{\rm{dim}}\,{\rm{Nul}}\,\,{A^T} &= 0.\end{aligned}\)

Step 3: Draw a conclusion

As \({\rm{dim}}\,{\rm{Nul}}\,\,{A^T} = 0\), the number of non-pivot columns in \({A^T}\) is 0, which implies that \({A^T}x = 0\) has a trivial solution.

Most popular questions for Math Textbooks

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.

In Exercise 17, A is an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if A is not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

Let \(A\) be an \(m \times n\) matrix of rank \(r > 0\) and let \(U\) be an echelon form of \(A\). Explain why there exists an invertible matrix \(E\) such that \(A = EU\), and use this factorization to write \(A\) as the sum of \(r\) rank 1 matrices. [Hint: See Theorem 10 in Section 2.4.]

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\) if \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\) and \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]\).
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