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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q1.2 23E (page 19)

How many types of 3脳2 matrices in reduced row-echelon form are there?

Short Answer

Expert verified

There are four type of 3脳2 matrices in reduced row-echelon.

Step by step solution

01

Form of 3×2 matrix

All type of 3脳2 matrix can be written as $[\begin{matrix} a & b \\ c & d \\ e & f \end{matrix}]$

Where a, b, c, d, e, f are constants.

02

3×2 matrix of reduced echelon form

In the above matrix if we want to reduce it to echelon form

Then we have three options for that

1. At the place of b we have 1. And all other values are 0.

$[\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$

2. At the place of a we have 1. And at place of b and f we can have y and x respectively, and all other values are 0.

$[\begin{matrix} 1 & y \\ 0 & 0 \\ 0 & x \end{matrix}]$

3. At place of a and d we have 1. And all other values are 0.

$[\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}]$

4. All the values are 0.

$[\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$

Hence, there are four types of 3脳2 matrices in reduced row-echelon.

$[\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 1 & y \\ 0 & 0 \\ 0 & x \end{matrix}], [\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$

Where x and y are constant.

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Most popular questions from this chapter

Consider a solution $\overset{鈫�}{x_{1}}$ of the linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ . Justify the facts stated in parts (a) and (b):

a. If ${\overset{鈫�}{x}}_{h}$ is a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A = \overset{鈫�}{x} = \overset{鈫�}{b}$ .

b. If $\overset{鈫�}{x_{2}}$ is another solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A \overset{鈫�}{x} + \overset{鈫�}{0}$ .

c. Now suppose A is a $2 脳 2$ matrix. A solution vector $\overset{鈫�}{x_{1}}$ of the system $A \overset{鈫�}{x} + \overset{鈫�}{b}$ is shown in the accompanying figure. We are told that the solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ form the line shown in the sketch. Draw the line consisting of all solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

If you are puzzled by the generality of this problem, think about an example first:

$A = (\begin{array}{ll} 1 鈥呪赌呪赌呪赌� 2 \\ 3 鈥呪赌呪赌呪赌� 6 \end{array}), \overset{鈫�}{b} = [\begin{array}{l} 3 \\ 9 \end{array}] and \overset{鈫�}{x_{1}} = [\begin{array}{l} 1 \\ 1 \end{array}]$

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product two ways: in terms of the columns of A and in terms of the rows of A.

13.

$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 7 \\ 11 \end{matrix}]$

Question:A linear system of the form $A \overset{鈫�}{x} = 0$ is called homogeneous. Justify the following facts:

a.All homogeneous systems are consistent.

b.A homogeneous system with fewer equations than unknowns has infinitely many solutions.

c.If $\overset{鈫�}{x_{1}} and \overset{鈫�}{x_{2}}$ are solutions of the homogeneous system $A \overset{鈫�}{x} = 0$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{2}}$ is a solution as well.

d.If $\overset{鈫�}{x}$ is a solution of the homogeneous system $A \overset{鈫�}{x} = 0$ andkis an arbitrary constant, then $k \overset{鈫�}{x}$ is a solution as well.

Let A be a 4 脳 4 matrix, and let $\overset{鈫�}{b}$ and $\overset{鈫�}{c}$ be two vectors in ${鈩�}^{4}$ . We are told that the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ has a unique solution. What can you say about the number of solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{c}$ ?

Consider a positive definite quadratic form q on $R^{n}$ with symmetric matrix. We know that there exists an orthonormal eigenbasis for ${\overset{鈫�}{v}}_{1}, . . ., {\overset{鈫�}{v}}_{n}$ for A, with associated positive eigenvalues $位_{1}, . . ., 位_{n}$ . Now consider the orthogonal Eigen basis ${\overset{鈫�}{w}}_{1}, 鈥�, {\overset{鈫�}{w}}_{n}$ , where ${\overset{鈫�}{w}}_{1} = \frac{1}{\sqrt{位}} {\overset{鈫�}{v}}_{1}$ .

Show that $q (c_{1} {\overset{鈫�}{w}}_{1} + . . . . . . + c_{n} {\overset{鈫�}{w}}_{n}) = c_{n}^{2}$ .

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