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Chapter 2: Q2.9-14Q (page 93)

In Exercises 13 and 14, find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?

14. \(\left[ {\begin{array}{*{20}{c}}1\\{ - {\bf{1}}}\\{ - 2}\\{\bf{5}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{3}}}\\{ - {\bf{1}}}\\{\bf{6}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{2}}\\{ - {\bf{6}}}\\{\bf{8}}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{4}}\\{ - {\bf{7}}}\\7\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}3\\{ - 8}\\9\\{ - 5}\end{array}} \right]\)

Short Answer

Expert verified

The basis is \(\left\{ {\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\{ - 2}\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\{ - 1}\\6\end{array}} \right]} \right\}\). The dimension is 2.

Step by step solution

01

Bases for Col A

The set of all linear combinations of the columns of matrix A is Col A, or it is called the column space of A. Pivot columns are the bases for Col A.

02

Obtain the row reduced echelon form

Consider the vectors\(\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\{ - 2}\\5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\{ - 1}\\6\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}0\\2\\{ - 6}\\8\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\4\\{ - 7}\\7\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}3\\{ - 8}\\9\\{ - 5}\end{array}} \right]\).

As it is given that four vectors span the column spaceof a matrix, soconstruct matrix A by using the given vectors as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}1&2&0&{ - 1}&3\\{ - 1}&{ - 3}&2&4&{ - 8}\\{ - 2}&{ - 1}&{ - 6}&{ - 7}&9\\5&6&8&7&{ - 5}\end{array}} \right]\)

Obtain the echelon form of matrix A as shown below:

Add row 1 to row 2 to get row 2.

\(A = \left[ {\begin{array}{*{20}{c}}1&2&0&{ - 1}&3\\0&{ - 1}&2&3&{ - 5}\\{ - 2}&{ - 1}&{ - 6}&{ - 7}&9\\5&6&8&7&{ - 5}\end{array}} \right]\)

Add 2 times row 1 to row 3 to get row 3.

\(A = \left[ {\begin{array}{*{20}{c}}1&2&0&{ - 1}&3\\0&{ - 1}&2&3&{ - 5}\\0&3&{ - 6}&{ - 9}&{15}\\5&6&8&7&{ - 5}\end{array}} \right]\)

Add\( - 5\)times row 1 to row 4 to get row 4.

\(A = \left[ {\begin{array}{*{20}{c}}1&2&0&{ - 1}&3\\0&{ - 1}&2&3&{ - 5}\\0&3&{ - 6}&{ - 9}&{15}\\0&{ - 4}&8&{12}&{ - 20}\end{array}} \right]\)

Add 3 times row 2 to row 3 to get row 3. Also, add\( - 4\)row 2 to row 4 to get row 4.

\(A = \left[ {\begin{array}{*{20}{c}}1&2&0&{ - 1}&3\\0&{ - 1}&2&3&{ - 5}\\0&0&0&0&0\\0&0&0&0&0\end{array}} \right]\)

03

Write the bases for Col A

To identify the pivot and the pivot position, observe the matrix鈥檚 leftmost column (nonzero column), which is the pivot column. At the top of this column, 1 is the pivot.

It is observed that the first and second columns have pivot elements.

The corresponding columns of matrix A are shown below:

\(\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\{ - 2}\\5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\{ - 1}\\6\end{array}} \right]\)

The column space is given as shown below:

\({\rm{Col }}A = \left\{ {\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\{ - 2}\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\{ - 1}\\6\end{array}} \right]} \right\}\)

Thus, the basis for Col A is \(\left\{ {\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\{ - 2}\\5\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\{ - 1}\\6\end{array}} \right]} \right\}\).

04

Dimension of basis

It is observed that matrix A has two pivot columns; so the dimension of basis A is 2.

Therefore, \({\rm{dim}}\left( A \right) = 2\).

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Most popular questions from this chapter

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation. Explain why T is both one-to-one and onto \({\mathbb{R}^n}\). Use equations (1) and (2). Then give a second explanation using one or more theorems.

Use partitioned matrices to prove by induction that for \(n = 2,3,...\), the \(n \times n\) matrices \(A\) shown below is invertible and \(B\) is its inverse.

\[A = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\1&1&0&{}&0\\1&1&1&{}&0\\ \vdots &{}&{}& \ddots &{}\\1&1&1& \ldots &1\end{array}} \right]\]

\[B = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\{ - 1}&1&0&{}&0\\0&{ - 1}&1&{}&0\\ \vdots &{}& \ddots & \ddots &{}\\0&{}& \ldots &{ - 1}&1\end{array}} \right]\]

For the induction step, assume A and Bare \(\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrices, and partition Aand B in a form similar to that displayed in Exercises 23.

Exercises 15 and 16 concern arbitrary matrices A, B, and Cfor which the indicated sums and products are defined. Mark each statement True or False. Justify each answer.

15. a. If A and B are \({\bf{2}} \times {\bf{2}}\) with columns \({{\bf{a}}_1},{{\bf{a}}_2}\) and \({{\bf{b}}_1},{{\bf{b}}_2}\) respectively, then \(AB = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}{{\bf{b}}_1}}&{{{\bf{a}}_2}{{\bf{b}}_2}}\end{aligned}} \right)\).

b. Each column of ABis a linear combination of the columns of Busing weights from the corresponding column of A.

c. \(AB + AC = A\left( {B + C} \right)\)

d. \({A^T} + {B^T} = {\left( {A + B} \right)^T}\)

e. The transpose of a product of matrices equals the product of their transposes in the same order.

Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?

When a deep space probe launched, corrections may be necessary to place the probe on a precisely calculated trajectory. Radio elementary provides a stream of vectors, \({{\bf{x}}_{\bf{1}}},....,{{\bf{x}}_k}\), giving information at different times about how the probe鈥檚 position compares with its planned trajectory. Let \({X_k}\) be the matrix \(\left[ {{x_{\bf{1}}}.....{x_k}} \right]\). The matrix \({G_k} = {X_k}X_k^T\) is computed as the radar data are analyzed. When \({x_{k + {\bf{1}}}}\) arrives, a new \({G_{k + {\bf{1}}}}\) must be computed. Since the data vector arrive at high speed, the computational burden could be serve. But partitioned matrix multiplication helps tremendously. Compute the column-row expansions of \({G_k}\) and \({G_{k + {\bf{1}}}}\) and describe what must be computed in order to update \({G_k}\) to \({G_{k + {\bf{1}}}}\).
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