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Chapter 1: Q1E (page 1)

Compute the products in Exercises 1鈥�4 using (a) the definition, as

in Example 1, and (b) the row鈥搗ector rule for computing \(A{\bf{x}}\). If a product is undefined, explain why.

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

Short Answer

Expert verified

The product is not defined because the number of columns in the matrix does not match the number of entries in the vector.

Step by step solution

01

Write the condition for the product of a vector and a matrix

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

Also, the product by using the row-vector rule is defined as shown below:

\(\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\ = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}\)

The number of columns in matrix \(A\) should be equal to the number of entries in vector x so that \(A{\bf{x}}\) can be defined.

02

Obtain the number of columns in matrix A

Consider matrix \(A = \left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]\).

It is observed that the number of columns in matrix \(A\) is 2.

03

Obtain the number of entries in vector x

Consider matrix \(x = \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]\).

It is observed that the number of entries in vector x is 3.

04

Check if \(Ax\) is defined or not

Since the number of columns in matrix \(A\) is not equal to the number of entries in vector x; so \(A{\bf{x}}\) cannot be defined.

Therefore, the product of \(\left[ {\begin{array}{*{20}{c}}{ - 4}&2\\1&6\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\7\end{array}} \right]\) is undefined.

Hence, the product is not defined.

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Most popular questions from this chapter

Determine which of the matrices in Exercises 7鈥�12areorthogonal. If orthogonal, find the inverse.

11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).

The solutions \(\left( {x,y,z} \right)\) of a single linear equation \(ax + by + cz = d\)

form a plane in \({\mathbb{R}^3}\) when a, b, and c are not all zero. Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have no points in common. Typical graphs are illustrated in the figure.

Three planes intersecting in a line.

(a)

Three planes intersecting in a point.

(b)

Three planes with no intersection.

(c)

Three planes with no intersection.

(肠鈥�)

Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and

\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).
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