Which of the following subsets of R3x3 are subspaces of R³x3 -0- 0 A. The 3 x 3 matrices A such that the vector B. The 3 x 3 matrices in reduced row-echelon form C. The 3 x 3 matrices of rank 1 is in the kernel of A
Which of the following subsets of R3x3 are subspaces of R³x3 -0- 0 A. The 3 x 3 matrices A such that the vector B. The 3 x 3 matrices in reduced row-echelon form C. The 3 x 3 matrices of rank 1 is in the kernel of A
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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![**Educational Content: Understanding Subspaces of \( \mathbb{R}^{3 \times 3} \)**
The task is to determine which of the following subsets of \( \mathbb{R}^{3 \times 3} \) are subspaces of \( \mathbb{R}^{3 \times 9} \):
1. **Option A:** The \( 3 \times 3 \) matrices \( A \) such that the vector
\[
\begin{pmatrix}
6 \\
0 \\
8
\end{pmatrix}
\]
is in the kernel of \( A \).
2. **Option B:** The \( 3 \times 3 \) matrices in reduced row-echelon form.
3. **Option C:** The \( 3 \times 3 \) matrices of rank 1.
4. **Option D:** The invertible \( 3 \times 3 \) matrices.
5. **Option E:** The \( 3 \times 3 \) matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries).
6. **Option F:** The \( 3 \times 3 \) matrices with all zeros in the second row.
**Explanation of Terms:**
- **Kernel of a Matrix:** A set of vectors that, when multiplied by the matrix, result in the zero vector.
- **Reduced Row-Echelon Form:** A form of a matrix where each leading entry (row's first nonzero number) is 1, and all elements in the column above and below a leading entry are zero.
- **Rank of a Matrix:** The dimension of the column space or the number of linearly independent columns or rows.
- **Invertible Matrices:** Matrices that have an inverse; only square matrices can be invertible and they must have full rank.
- **Trace of a Matrix:** The sum of elements on the main diagonal of the matrix.
**Understanding Subspaces:**
A subspace is a set of vectors in a vector space that satisfies three properties:
1. Contains the zero vector.
2. Closed under vector addition.
3. Closed under scalar multiplication.
Students are encouraged to examine each option to determine if they meet the criteria to be considered subspaces of \( \mathbb{R}^{3 \times 9} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feca0682c-5299-46ec-a99f-9ad1cb25da66%2F58de1649-06a4-4a6e-9409-66f67837a15a%2F682b96_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Understanding Subspaces of \( \mathbb{R}^{3 \times 3} \)**
The task is to determine which of the following subsets of \( \mathbb{R}^{3 \times 3} \) are subspaces of \( \mathbb{R}^{3 \times 9} \):
1. **Option A:** The \( 3 \times 3 \) matrices \( A \) such that the vector
\[
\begin{pmatrix}
6 \\
0 \\
8
\end{pmatrix}
\]
is in the kernel of \( A \).
2. **Option B:** The \( 3 \times 3 \) matrices in reduced row-echelon form.
3. **Option C:** The \( 3 \times 3 \) matrices of rank 1.
4. **Option D:** The invertible \( 3 \times 3 \) matrices.
5. **Option E:** The \( 3 \times 3 \) matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries).
6. **Option F:** The \( 3 \times 3 \) matrices with all zeros in the second row.
**Explanation of Terms:**
- **Kernel of a Matrix:** A set of vectors that, when multiplied by the matrix, result in the zero vector.
- **Reduced Row-Echelon Form:** A form of a matrix where each leading entry (row's first nonzero number) is 1, and all elements in the column above and below a leading entry are zero.
- **Rank of a Matrix:** The dimension of the column space or the number of linearly independent columns or rows.
- **Invertible Matrices:** Matrices that have an inverse; only square matrices can be invertible and they must have full rank.
- **Trace of a Matrix:** The sum of elements on the main diagonal of the matrix.
**Understanding Subspaces:**
A subspace is a set of vectors in a vector space that satisfies three properties:
1. Contains the zero vector.
2. Closed under vector addition.
3. Closed under scalar multiplication.
Students are encouraged to examine each option to determine if they meet the criteria to be considered subspaces of \( \mathbb{R}^{3 \times 9} \).
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