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Chapter 2: Problem 33

In Exercises 31鈥�36, respond as comprehensively as possible, and justify your answer. If \(Q\) is a \(4 \times 4\) matrix and \(\mathrm{Col} Q=\mathbb{R}^{4},\) what can you say about solutions of equations of the form \(Q \mathbf{x}=\mathbf{b}\) for \(\mathbf{b}\) in \(\mathbb{R}^{4} ?\)

Short Answer

Expert verified
The equation \(Q\mathbf{x} = \mathbf{b}\) has a unique solution for any \(\mathbf{b}\) in \(\mathbb{R}^4\).

Step by step solution

01

Understand the Given Information

The problem involves a \(4 \times 4\) matrix \(Q\). We are given that the column space \(\text{Col} Q\) is equal to \(\mathbb{R}^4\). This means the columns of \(Q\) span the entire \(\mathbb{R}^4\) space.
02

Interpret the Column Space Condition

Since \(\text{Col} Q = \mathbb{R}^4\), the columns of \(Q\) must form a basis for \(\mathbb{R}^4\). Therefore, they are linearly independent. A \(4 \times 4\) matrix with linearly independent columns is invertible.
03

Determine the Implication for \(Q\)

An invertible matrix \(Q\) implies that it has full rank, which is 4 in the case of a \(4 \times 4\) matrix. This ensures that the system \(Q\mathbf{x} = \mathbf{b}\) has a unique solution for any vector \(\mathbf{b}\) in \(\mathbb{R}^4\).
04

Conclude the Solution Existence and Uniqueness

Since \(Q\) is invertible, for every \(\mathbf{b}\) in \(\mathbb{R}^4\), there exists exactly one \(\mathbf{x}\) such that \(Q\mathbf{x} = \mathbf{b}\). Therefore, given \(Q\mathbf{x} = \mathbf{b}\), a unique solution exists for every \(\mathbf{b}\) in \(\mathbb{R}^4\).

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Key Concepts

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

Column Space
The column space of a matrix, often denoted as \(\text{Col} \ Q\), is one of its fundamental elements. It refers to the set of all possible vectors that can be obtained by taking linear combinations of the matrix's columns. For a \(4 \times 4\) matrix \(Q\), if \(\text{Col} \ Q = \mathbb{R}^4\), this implies that the columns of \(Q\) span the entire four-dimensional space \(\mathbb{R}^4\). In simpler terms, any vector in \(\mathbb{R}^4\) can be expressed as a combination of the columns of \(Q\). This is a crucial property because it lays the groundwork for understanding the invertibility of the matrix and the behavior of the solutions to matrix equations.
Linear Independence
Linear independence is a key concept when analyzing matrices. If a set of vectors is linearly independent, no vector in the set can be written as a linear combination of the others.
  • Implication for Matrix: In our context with a \(4 \times 4\) matrix \(Q\), if the column space equals \(\mathbb{R}^4\), it implies the columns of \(Q\) are linearly independent.
  • Convertibility: This linear independence is essential for the matrix to be invertible. An invertible matrix confirms that it does not "squash" dimensions, allowing one-to-one mappings.
By understanding that \(Q\) has linearly independent columns, you gain insight into why the solutions to equations involving \(Q\) have specific properties, such as uniqueness.
Unique Solution
When you solve an equation like \(Q \mathbf{x} = \mathbf{b}\), where \(Q\) is a \(4 \times 4\) invertible matrix, an important outcome is the uniqueness of the solution.
  • Why It Matters: With such a matrix, the solution vector \(\mathbf{x}\) that satisfies the equation for any vector \(\mathbf{b}\) in \(\mathbb{R}^4\) is unique.
  • Role of Invertibility: The matrix's invertibility ensures that there are no multiple solutions; for each \(\mathbf{b}\), only one \(\mathbf{x}\) can solve the equation.
Understanding the uniqueness of solutions gives clarity in applications like solving systems of equations where precise outcomes are necessary.
Full Rank
The term "full rank" refers to the rank of a matrix being as large as possible for matrices of its dimension. For a \(4 \times 4\) matrix like \(Q\), having full rank means it has a rank of 4.
  • Significance: Full rank indicates that all columns (or rows, equivalently) are linearly independent and span the space completely.
  • Consequence for Solutions: A matrix with full rank implies that it is invertible, guaranteeing unique solutions for equations of the form \(Q\mathbf{x} = \mathbf{b}\).
In essence, when a matrix is of full rank, it assures you that none of the columns are redundant, and the matrix is as effective as it can be in solving equations uniquely in its dimensional space.

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Most popular questions from this chapter

Suppose \(T\) and \(U\) are linear transformations from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\) such that \(T(U \mathbf{x})=\mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n} .\) Is it true that \(U(T \mathbf{x})=\mathbf{x}\) for all \(\mathbf{x}\) in \(\mathbb{R}^{n}\) ? Why or why not?

Suppose a \(4 \times 7\) matrix \(A\) has three pivot columns. Is Col \(A=\mathbb{R}^{3} ?\) What is the dimension of Nul \(A ?\) Explain your answers.

Let \(S\) be the triangle with vertices \((9,3,-5),(12,8,2),\) \((1.8,2.7,1) .\) Find the image of \(S\) under the perspective projection with center of projection at \((0,0,10)\)

Suppose a \(3 \times 5\) matrix \(A\) has three pivot columns. Is \(\mathrm{Col}\) \(A=\mathbb{R}^{3} ?\) Is \(\mathrm{Nul} A=\mathbb{R}^{2} ?\) Explain your answers.

In Exercises 21 and \(22,\) mark each statement True or False. Justify each answer. a. A subspace of \(\mathbb{R}^{n}\) is any set \(H\) such that (i) the zero vector is in \(H,\) (ii) \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u}+\mathbf{v}\) are in \(H,\) and (iii) \(c\) is a scalar and \(c \mathbf{u}\) is in \(H .\) b. If \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\) are in \(\mathbb{R}^{n},\) then \(\operatorname{Span}\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\\}\) is the same as the column space of the matrix \(\left[\mathbf{v}_{1} \cdots \mathbf{v}_{p}\right] .\) c. The set of all solutions of a system of \(m\) homogeneous equations in \(n\) unknowns is a subspace of \(\mathbb{R}^{m} .\) d. The columns of an invertible \(n \times n\) matrix form a basis for \(\mathbb{R}^{n}\) . e. Row operations do not affect linear dependence relations among the columns of a matrix.
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