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Chapter 1: Q28E (page 1)

How many pivot columns must a \(5 \times 7\) matrix have if its columns span \({\mathbb{R}^5}\)? Why?

Short Answer

Expert verified

The \(5 \times 7\) matrix must have five pivot columns.

Step by step solution

01

Determine the pivot position if the columns span \({\mathbb{R}^5}\)

Theorem 4 states that\(A\)is an\({\mathop{\rm m}\nolimits} \times n\)matrix. For this, the following statements are equivalent.

  1. For each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\), the equation \(Ax = b\) has a solution.
  2. Each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\) is a linear combination of the columns of A.
  3. The columns of \(A\) span .
  4. \(A\)has a pivot position in every row.

According to theorem 4, if the columns of a \(5 \times 7\) matrix \(A\) span \({\mathbb{R}^5}\), matrix \(A\) has a pivot in every row.

02

Determine the number of pivot columns in the \(5 \times 7\) matrix

Matrix \(A\) contains five pivot columns since each pivot position is in a different column.

Thus, the \(5 \times 7\) matrix must have five pivot columns.

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Most popular questions from this chapter

Find the general solutions of the systems whose augmented matrices are given

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase “columns of A.”
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)
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