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Chapter 2: Problem 51

a. Consider an \(n \times m\) matrix \(A\) with \(\operatorname{rank}(A)m .\) Show that there exists a vector \(\vec{b}\) in \(\mathbb{R}^{n}\) such that the system \(A \vec{x}=\vec{b}\) is inconsistent.

Short Answer

Expert verified
For part a: There exists a vector \(\vec{b}\) in \(\mathbb{R}^n\) making the system \(A\vec{x} = \vec{b}\) inconsistent due to the rank of A being less than n. For part b: Since n>m, the rank of matrix A is insufficient to span \(\mathbb{R}^n\), guaranteeing the existence of an inconsistent \(\vec{b}\).

Step by step solution

01

Understand the effect of matrix rank on solutions

The rank of a matrix is the dimension of the vector space spanned by its rows (row rank) or columns (column rank). If the rank of an n x m matrix A is less than n, it means that the rows of A do not span all of \(\rbb{R^n}\). Therefore, there exist vectors in \(\mathbb{R}^n\) that cannot be expressed as linear combinations of the rows of A, which implies the existence of a vector \(\vec{b}\) in \(\mathbb{R}^n\) making the system inconsistent.
02

Applying the rref form of the matrix

Let E be the reduced row echelon form (rref) of A. Since E has the same rank as A and \(\operatorname{rank}(A) < n\), it follows that E will have at least one row of zeros. This row of zeros corresponds to an equation that cannot be satisfied by any vector, which implies there is a vector \(\vec{c}\) in \(\mathbb{R}^n\) such that the system \(E \vec{x} = \vec{c}\) is inconsistent. By choosing \(\vec{c}\) to have a non-zero entry in the position corresponding to the row of zeros in E, we ensure that the equation represented by that row is not satisfied.
03

Constructing the inconsistent \(\vec{b}\) for matrix A

We know that the operations that convert A to its rref E can be represented by an invertible matrix T such that \(E = TA\). Therefore, the system \(E \vec{x} = \vec{c}\) is equivalent to \(TA \vec{x} = \vec{c}\). Since this system is inconsistent, we can choose \(\vec{b} = T^{-1}\vec{c}\) to create an inconsistent system with the original matrix A. The system \(A \vec{x} = \vec{b}\) will then be inconsistent, because multiplying both sides by T gives the inconsistent system \(TA\vec{x} = \vec{c}\) back.
04

Addressing the case where \(n>m\)

If an n x m matrix A has \(n>m\), then the maximum rank A can have is m, which is less than n. Following the logic from the previous steps, since the rank of A is less than n, there must exist a vector \(\vec{b}\) in \(\mathbb{R}^n\) such that no \(\vec{x}\) can solve the system \(A\vec{x} = \vec{b}\). Hence, the system is inconsistent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Algebra
In the world of mathematics, linear algebra is a cornerstone that deals with vectors, vector spaces, matrices, and the algebraic structure that comes with linear equations and transformations. It's like having a toolbox where every tool is designed for operations involving planes and lines. Imagine linear algebra as the language of straight lines and flat surfaces, where everything we measure or calculate follows predictable, straight paths.
Reduced Row Echelon Form
The reduced row echelon form (rref) is a particular matrix format that simplifies solving systems of linear equations. To transform a matrix to its rref, we perform a series of elementary row operations, aiming to achieve a few goals: each leading coefficient (the first non-zero number from the left, in a non-zero row) is 1, each leading coefficient is the only non-zero number in its column, and the rows with all zero elements are at the bottom of the matrix. It's like organizing a messy room so you can easily see what you have and find what you need.
System of Linear Equations
A system of linear equations is a collection of one or more linear equations involving the same variables鈥攖hink of it as a quest to find a common solution that satisfies all equations at once. In the context of matrices, solving these systems is equivalent to finding a vector that, when multiplied by a given matrix, results in a particular solution vector.
Vector Space
A vector space is a set of elements called vectors, which can be scaled and added together following certain rules that make them a linear space. For example, imagine a three-dimensional space where any point can be reached by moving in three perpendicular directions. In linear algebra, a vector space includes all the possible points (vectors) we can get to using the basic components, or basis, of that space.
Rank of a Matrix
The rank of a matrix is the number of linearly independent rows or columns in the matrix. This is an important concept because it tells us about the matrix's ability to solve linear systems鈥攖he higher the rank, the more 'solving power' it holds. A full rank matrix can potentially offer a unique solution to a linear system, while a matrix with rank less than the number of its rows might not be able to provide a solution for certain target vectors, which brings us to the idea of inconsistency in systems.
Inconsistent System
An inconsistent system is one where there's no solution that satisfies all the equations at the same time. In our matrix scenario, this happens when the set of linear transforms defined by the matrix cannot possibly reach a certain vector, meaning there's no vector that, when transformed, will land on the given target. It's like being given a map that lacks a path to the treasure; no matter how much you search, you won't find a way to the X that marks the spot.

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Most popular questions from this chapter

Consider an invertible \(n \times n\) matrix \(A\) and an \(n \times n\) matrix \(B\). A certain sequence of elementary row operations transforms \(A\) into \(I_{n}\) a. What do you get when you apply the same row operations in the same order to the matrix \(A B ?\) b. What do you get when you apply the same row operations to \(I_{n} ?\)

Decide whether the linear transformations are invertible. Find the inverse transformation if it exists. Do the computations with paper and pencil. Show all your work. $$\begin{array}{l} y_{1}=x_{1}+x_{2}+x_{3} \\ y_{2}=x_{1}+2 x_{2}+3 x_{3} \\ y_{3}=x_{1}+4 x_{2}+9 x_{3} \end{array}$$

Which of the functions \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) are invertible? $$f(x)=2^{x}$$

For two invertible \(n \times n\) matrices \(A\) and \(B,\) determine which of the formulas stated are necessarily true. \(A^{-1} B\) is invertible, and \(\left(A^{-1} B\right)^{-1}=B^{-1} A\)

a. Find the scaling matrix \(A\) that transforms \(\left[\begin{array}{r}2 \\\ -1\end{array}\right]\) into \(\left[\begin{array}{r}8 \\ -4\end{array}\right]\) b. Find the orthogonal projection matrix \(B\) that transforms \(\left[\begin{array}{l}2 \\ 3\end{array}\right]\) into \(\left[\begin{array}{l}2 \\ 0\end{array}\right]\) c. Find the rotation matrix \(C\) that transforms \(\left[\begin{array}{l}0 \\\ 5\end{array}\right]\) into \(\left[\begin{array}{l}3 \\ 4\end{array}\right]\) d. Find the shear matrix \(D\) that transforms \(\left[\begin{array}{l}1 \\\ 3\end{array}\right]\) into \(\left[\begin{array}{l}7 \\ 3\end{array}\right]\) e. Find the reflection matrix \(E\) that transforms \(\left[\begin{array}{l}7 \\\ 1\end{array}\right]\) into \(\left[\begin{array}{r}-5 \\ 5\end{array}\right]\)
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