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In Exercises 11 and 12, the matrices are all n n. Each part of the exercises is an implication of the form 鈥淚f 鈥渟tatement 1鈥�, then 鈥渟tatement 2鈥�.鈥� Mark an implication as True if the truth of 鈥渟tatement 2鈥� always follows whenever 鈥渟tatement 1鈥� happens to be true. An implication is False if there is an instance in which 鈥渟tatement 2鈥� is false but 鈥渟tatement 1鈥� is true. Justify each answer. a. If there is an \(n \times n\) matrix \(D\) such that \(A D=I\) , then there is also an \(n \times n\) matrix \(C\) such that \(C A=I .\) b. If the columns of \(A\) are linearly independent, then the columns of \(A\) span \(\mathbb{R}^{n}\) . c. If the equation \(A \mathbf{x}=\mathbf{b}\) has at least one solution for each \(\mathbf{b}\) in \(\mathbb{R}^{n},\) then the solution is unique for each \(\mathbf{b} .\) d. If the linear transformation \((\mathrm{x}) \mapsto A \mathbf{x}\) maps \(\mathbb{R}^{n}\) into \(\mathbb{R}^{n}\) then \(A\) has \(n\) pivot positions. e. If there is a \(\mathbf{b}\) in \(\mathbb{R}^{n}\) such that the equation \(A \mathbf{x}=\mathbf{b}\) is inconsistent, then the transformation \(\mathbf{x} \mapsto A \mathbf{x}\) is not one- to-one.

Step by step solution

01

Understanding Statement a

We are given that there is an \(n \times n\) matrix \(D\) such that \(AD = I\). This implies that \(A\) has a right inverse, which, for square matrices, also implies that \(A\) is invertible. If \(A\) is invertible, then there exists a matrix \(C\) such that \(CA = I\), i.e., \(C = A^{-1}\). Therefore, this statement is True.

02

Evaluating Statement b

The columns of a matrix \(A\) being linearly independent implies that \(A\) is an invertible matrix. An \(n \times n\) invertible matrix has a rank equal to \(n\), which means its column vectors span \(\mathbb{R}^{n}\). Hence, this statement is True.

03

Analyzing Statement c

If the equation \(A \mathbf{x}=\mathbf{b}\) has at least one solution for each \(\mathbf{b}\) in \(\mathbb{R}^{n}\), it indicates that \(A\) is onto. However, onto does not imply unique solutions unless \(A\) is invertible. Hence, without further information about \(A\), the uniqueness is not confirmed. Thus, this statement is False.

04

Checking Statement d

A linear transformation \(\mathbf{x} \mapsto A \mathbf{x}\) mapping \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\) just indicates a function with domain and co-domain in \(\mathbb{R}^{n}\), not necessarily surjective or injective. Without additional information, we cannot conclude \(A\) has \(n\) pivot positions purely based on these mapping properties. Therefore, the statement is False.

05

Verifying Statement e

If there exists a \(\mathbf{b}\) such that the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent, then the system has no solution for that \(\mathbf{b}\), showing \(A\) does not map onto \(\mathbb{R}^{n}\), indicating that \(\mathbf{x} \mapsto A \mathbf{x}\) is not one-to-one. This statement is True.

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Key Concepts

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

Invertible Matrices

Invertible matrices are a fundamental concept in linear algebra, closely related to identity matrices. A square matrix \(A\) is invertible if there exists another matrix \(B\) of the same size such that the product of \(A\) and \(B\) is the identity matrix. This is mathematically denoted as \(A \cdot B = I\) and \(B \cdot A = I\). Here, \(I\) is the identity matrix with ones on the diagonal and zeros elsewhere.

• An invertible matrix is also called a non-singular or a non-degenerate matrix.
• The inverse of a matrix \(A\), often denoted as \(A^{-1}\), is unique.
• Not all matrices are invertible. A matrix must be square (same number of rows and columns) and have full rank to be invertible.

Understanding the concept of invertibility is crucial because it implies that the transformation associated with the matrix has an exact "undo" operation, represented by its inverse matrix. This makes invertible matrices useful in solving linear equations and analyzing linear transformations.

Linear Independence

Linear independence is a key concept used to understand the structure of vector spaces. A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. In simpler terms, no vector is redundant; each vector adds a new direction in space.

• If the columns of a matrix \(A\) are linearly independent, then \(A\) is full-rank, meaning it has maximum rank, equal to the number of its rows or columns.
• This property is crucial when determining whether a matrix is invertible.
• For a matrix with linearly independent columns, any linear combination that equals the zero vector must have all coefficients equal to zero.

The importance of linear independence lies in its ability to tell us about the "spanning" capabilities of vector sets. In terms of the columns of a matrix, linear independence signifies that they cover the whole space without overlap, and thus are capable of spanning \(\mathbb{R}^{n}\) when \(n\) is the number of columns.

Pivot Positions

Pivot positions are specific locations in a matrix that help in understanding its rank and its solution properties when forming equations. They are the first non-zero entries in rows of a matrix that has been transformed into row-echelon form through Gaussian elimination.

• Each pivot position is associated with a pivot column. These pivot columns determine the basic variables in a solution of a linear system.
• The number of pivot positions in a matrix is equal to its rank. Thus, a full-rank matrix (invertible) will have a pivot position in each row/column.
• Recognizing pivot positions in a matrix helps in understanding linear transformations, as they show the dimensions of the image of the matrix.

By identifying pivot positions, one can determine the solvability of a system of linear equations and analyze whether a matrix transformation is one-to-one or onto. Pivot positions hence play a critical role in simplifying complex linear algebra problems into manageable steps.