2. Let A = [1000] 1 1 00 01 1 0 001 1 Answer the following questions. a) Do the columns of A span R*? Why? b) Are the columns of A linearly dependent? Why? c) How many solutions has an equation Ax = b, where b = d) Is the linear transformation T:x→ Ax invertible? Why? ? Why?
2. Let A = [1000] 1 1 00 01 1 0 001 1 Answer the following questions. a) Do the columns of A span R*? Why? b) Are the columns of A linearly dependent? Why? c) How many solutions has an equation Ax = b, where b = d) Is the linear transformation T:x→ Ax invertible? Why? ? Why?
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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Question
![**Matrix Analysis and Linear Algebra**
Consider the matrix \( A \):
\[
A = \begin{bmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
\end{bmatrix}
\]
**2. Let \( A \) be as defined above. Answer the following questions:**
**a) Do the columns of \( A \) span \(\mathbb{R}^4\)? Why?**
The columns of \( A \) span \(\mathbb{R}^4\) if they can form a basis for \(\mathbb{R}^4\), meaning they are linearly independent and there are four vectors.
**b) Are the columns of \( A \) linearly dependent? Why?**
The columns of \( A \) are linearly dependent if there exist scalar coefficients, not all zero, such that a linear combination of the columns results in the zero vector.
**c) How many solutions does the equation \( A\mathbf{x} = \mathbf{b} \) have, where \(\mathbf{b} = \begin{bmatrix} -2 \\ 1 \\ 7 \\ 3 \end{bmatrix}\)? Why?**
To determine the number of solutions, analyze the consistency of the system and the rank of matrix \( A \) relative to the augmented matrix \( [A|\mathbf{b}] \).
**d) Is the linear transformation \( T: \mathbf{x} \mapsto A\mathbf{x} \) invertible? Why?**
The linear transformation \( T \) is invertible if \( A \) is a square matrix with linearly independent columns, ensuring that \( A \) has a nonzero determinant.
Explanation of Diagrams/Graphs (if any): The image contains a visual representation of a 4x4 matrix \( A \) and a column vector \( \mathbf{b} \). The matrix is displayed to help solve the linear algebra problems presented in questions (a) through (d). There are no additional graphs or charts present in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff64dcbf8-f12f-4842-986c-dba4c32e364d%2F5dd174df-dfcc-495b-b148-04eb8854602e%2Fuswurcq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Analysis and Linear Algebra**
Consider the matrix \( A \):
\[
A = \begin{bmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
\end{bmatrix}
\]
**2. Let \( A \) be as defined above. Answer the following questions:**
**a) Do the columns of \( A \) span \(\mathbb{R}^4\)? Why?**
The columns of \( A \) span \(\mathbb{R}^4\) if they can form a basis for \(\mathbb{R}^4\), meaning they are linearly independent and there are four vectors.
**b) Are the columns of \( A \) linearly dependent? Why?**
The columns of \( A \) are linearly dependent if there exist scalar coefficients, not all zero, such that a linear combination of the columns results in the zero vector.
**c) How many solutions does the equation \( A\mathbf{x} = \mathbf{b} \) have, where \(\mathbf{b} = \begin{bmatrix} -2 \\ 1 \\ 7 \\ 3 \end{bmatrix}\)? Why?**
To determine the number of solutions, analyze the consistency of the system and the rank of matrix \( A \) relative to the augmented matrix \( [A|\mathbf{b}] \).
**d) Is the linear transformation \( T: \mathbf{x} \mapsto A\mathbf{x} \) invertible? Why?**
The linear transformation \( T \) is invertible if \( A \) is a square matrix with linearly independent columns, ensuring that \( A \) has a nonzero determinant.
Explanation of Diagrams/Graphs (if any): The image contains a visual representation of a 4x4 matrix \( A \) and a column vector \( \mathbf{b} \). The matrix is displayed to help solve the linear algebra problems presented in questions (a) through (d). There are no additional graphs or charts present in the image.
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