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Check the true statements below: A. A basis is a spanning set that is as large as possible. B. The columns of an invertible 1 x matrix form a basis for R", C. I H = span{b₁,...,b), then (b₁,...,bo) is a basis for H. D. A single vector by itself is linearly dependent. E. In some cases, the linear dependence relations amoung the columns of a matrix can be affected by certain elementary row operations on the matrix.

Check the true statements below: A. A basis is a spanning set that is as large as possible. B. The columns of an invertible 1 x matrix form a basis for R", C. I H = span{b₁,...,b), then (b₁,...,bo) is a basis for H. D. A single vector by itself is linearly dependent. E. In some cases, the linear dependence relations amoung the columns of a matrix can be affected by certain elementary row operations on the matrix.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Understanding Basis and Linear Dependence in Matrices** **Objective: Identify the True Statements** 1. **Statement A:** A basis is a spanning set that is as large as possible. 2. **Statement B:** The columns of an invertible \( n \times n \) matrix form a basis for \(\mathbb{R}^n\). 3. **Statement C:** If \( H = \text{span}\{ \mathbf{b}_1, \ldots, \mathbf{b}_j \} \), then \(\{ \mathbf{b}_1, \ldots, \mathbf{b}_j \}\) is a basis for \( H \). 4. **Statement D:** A single vector by itself is linearly dependent. 5. **Statement E:** In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix. --- **Explanation of Concepts:** - **Basis:** In linear algebra, a basis of a vector space is a set of vectors that are linearly independent and span the entire space. - **Spanning Set:** A set of vectors span a vector space if any vector in the space can be expressed as a linear combination of the set. - **Linear Independence:** A set of vectors is linearly independent if no vector in the set can be expressed as a combination of the others. - **Invertible Matrix:** An \( n \times n \) matrix is invertible if there exists an \( n \times n \) matrix such that their product is the identity matrix. - **Elementary Row Operations:** Operations including row swapping, scaling rows, and adding multiples of one row to another, which can alter the matrix properties concerning linear dependence.
Transcribed Image Text:**Title: Understanding Basis and Linear Dependence in Matrices** **Objective: Identify the True Statements** 1. **Statement A:** A basis is a spanning set that is as large as possible. 2. **Statement B:** The columns of an invertible \( n \times n \) matrix form a basis for \(\mathbb{R}^n\). 3. **Statement C:** If \( H = \text{span}\{ \mathbf{b}_1, \ldots, \mathbf{b}_j \} \), then \(\{ \mathbf{b}_1, \ldots, \mathbf{b}_j \}\) is a basis for \( H \). 4. **Statement D:** A single vector by itself is linearly dependent. 5. **Statement E:** In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix. --- **Explanation of Concepts:** - **Basis:** In linear algebra, a basis of a vector space is a set of vectors that are linearly independent and span the entire space. - **Spanning Set:** A set of vectors span a vector space if any vector in the space can be expressed as a linear combination of the set. - **Linear Independence:** A set of vectors is linearly independent if no vector in the set can be expressed as a combination of the others. - **Invertible Matrix:** An \( n \times n \) matrix is invertible if there exists an \( n \times n \) matrix such that their product is the identity matrix. - **Elementary Row Operations:** Operations including row swapping, scaling rows, and adding multiples of one row to another, which can alter the matrix properties concerning linear dependence.
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