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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 the equation \(A \mathbf{x}=\mathbf{0}\) has only the trivial solution, then \(A\) is row equivalent to the \(n \times n\) identity matrix. b. If the columns of \(A\) span \(\mathbb{R}^{n}\) , then the columns are linearly independent. c. If \(A\) is an \(n \times n\) matrix, then the equation \(A \mathbf{x}=\mathbf{b}\) has at least one solution for each \(\mathbf{b}\) in \(\mathbb{R}^{n} .\) d. If the equation \(A \mathbf{x}=\mathbf{0}\) has a nontrivial solution, then \(A\) has fewer than \(n\) pivot positions. e. If \(A^{T}\) is not invertible, then \(A\) is not invertible.

Step by step solution

01

Identify Assumptions and Results

We begin with the first statement: "If the equation \(A \mathbf{x} = \mathbf{0}\) has only the trivial solution, then \(A\) is row equivalent to the \(n \times n\) identity matrix." Here, the assumption is about the existence of only a trivial solution (\(\mathbf{x} = \mathbf{0}\)), and the result is the row equivalency to the identity matrix.

02

Evaluate Row Equivalence

If \(A \mathbf{x} = \mathbf{0}\) has only the trivial solution, this implies that \(A\) is invertible. An invertible matrix is row equivalent to the identity matrix, thus the statement is True. Row equivalency to the identity matrix implies full rank and hence invertibility.

03

Determine Column Span and Independence

For part (b), the statement is: "If the columns of \(A\) span \(\mathbb{R}^{n}\), then the columns are linearly independent." When columns span \(\mathbb{R}^{n}\), it means the matrix is of full rank, hence the columns must be linearly independent. Thus, the statement is True.

04

Check Existence of Solutions

Part (c) states: "If \(A\) is an \(n \times n\) matrix, then the equation \(A \mathbf{x} = \mathbf{b}\) has at least one solution for each \(\mathbf{b}\) in \(\mathbb{R}^{n}\)." For any vector \(\mathbf{b}\), having a solution requires \(A\) to be invertible or to have full rank; otherwise, solutions might not exist for some \(\mathbf{b}\). This statement is not inherently true unless further information about \(A\)'s invertibility is provided.

05

Consider Nontrivial Solutions and Pivot Positions

For part (d), "If the equation \(A \mathbf{x} = \mathbf{0}\) has a nontrivial solution, then \(A\) has fewer than \(n\) pivot positions." A nontrivial solution implies linear dependence among columns and thus fewer pivot positions than \(n\). Hence, the statement is True.

06

Analyze Invertibility of Transpose and Original

For part (e), "If \(A^{T}\) is not invertible, then \(A\) is not invertible." Since the invertibility of a matrix is equivalent to the invertibility of its transpose, if \(A^{T}\) is not invertible, then \(A\) cannot be invertible. Thus, this statement is True.

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Key Concepts

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

Row Equivalence

Row equivalence in matrix theory means that you can convert two different matrices into one another through a sequence of row operations. In practice, these operations include row swapping, scaling rows by non-zero constants, and adding or subtracting rows from one another.

If a matrix is row equivalent to the identity matrix, this tells us that the matrix is invertible. This is because only matrices of full rank can be transformed into an identity matrix through row operations.

• Full rank implies that every row (or column) has pivotal positions or leading ones.
• An identity matrix is a matrix that does not change any vector upon multiplication, so converting any matrix into an identity matrix inherently means it still has all the necessary information.

Hence, a matrix with only trivial solutions is indeed row equivalent to the identity matrix, confirming its invertibility.

Linear Independence

Understanding linear independence involves a set of vectors where no vector in the set is a linear combination of the others. If the columns of a matrix span the space, like \( \mathbb{R}^{n} \), they essentially fill the entire space, meaning that any vector can be written as a combination of these columns.

Linear independence ensures that:

• No column is superfluous; each one adds new information.
• The determinant of the matrix is non-zero, which aligns with invertibility.

Thus, when columns span \( \mathbb{R}^{n} \), they need to be independent. If they were not independent, they couldn't represent every possible vector in \( \mathbb{R}^{n} \).

Invertible Matrix

An invertible matrix is one that has an inverse, meaning there exists another matrix that can be multiplied with the original to yield the identity matrix. For square matrices, this happens when the matrix is of full rank; every row and column must be linearly independent.

Key properties of invertible matrices include:

• They must have determinant values not equal to zero, ensuring that the matrix operations preserve value.
• An invertible matrix implies that the system of linear equations it represents has unique solutions.

This characteristic of invertibility is crucial in applications across various domains like computer graphics and systems of equations where unique results are necessary.

Nontrivial Solution

In the context of equations involving matrices, a nontrivial solution exists when the equation \( A \mathbf{x} = \mathbf{0} \) is satisfied by any vector \( \mathbf{x} \) other than the zero vector. Nontrivial solutions indicate linear dependence among columns.

Considerations when nontrivial solutions are present include:

• The matrix has fewer pivot positions than its dimension, signifying less than full rank.
• It points to redundancies in the columns, suggesting some columns can be expressed as combinations of others.

Nontrivial solutions are particularly important in determining the rank of a matrix and structural composition of vector spaces.

Pivot Positions

Pivot positions are essential indicators of the rank and structure of a matrix. These are specific locations in rows turned into leading ones as a result of row operations during the process of Gaussian elimination. The concept is fundamental in both row equivalence and matrix rank analyzation.

Important aspects of pivot positions include:

• A matrix with pivot positions in every row indicates full rank, crucial for invertibility.
• In non-square matrices, pivot positions help confirm whether the system of equations is consistent.

Understanding pivot positions can simplify a matrix into reduced row-echelon form, highlighting its critical characteristics and applications in solving linear systems.