Mark the following statements true or false. If false, give an explanation of why it is false. (a) R² is a subspace of R³. (b) The column space of A is the range of the mapping x → Ax. (c) The kernel of a linear transformation is a vector space.
Mark the following statements true or false. If false, give an explanation of why it is false. (a) R² is a subspace of R³. (b) The column space of A is the range of the mapping x → Ax. (c) The kernel of a linear transformation is a vector space.
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:10. Mark the following statements true or false. If false, give an explanation of why it is false.
(a) \(\mathbb{R}^2\) is a subspace of \(\mathbb{R}^3\).
(b) The column space of \(A\) is the range of the mapping \(x \rightarrow Ax\).
(c) The kernel of a linear transformation is a vector space.
(d) A single vector by itself is linearly dependent.
(e) If \(H = \text{Span}\{ \mathbf{b}_1, \ldots, \mathbf{b}_p \}\), then \(\{ \mathbf{b}_1, \ldots, \mathbf{b}_p \}\) is a basis for \(H\).
(f) The columns of an invertible \(n \times n\) matrix form a basis for \(\mathbb{R}^n\).
(g) If \(f\) is a function in the vector space \(V\) of all real-valued functions on \(\mathbb{R}\) and if \(f(t) = 0\) for some \(t\), then \(f\) is the zero vector in \(V\).
(h) If a finite set \(S\) of nonzero vectors spans a vector space \(V\), then some subset of \(S\) is a basis for \(V\).
![**Problem 9: Linear Algebra - Finding a Basis**
**Objective:**
Find a basis for the space spanned by the given vectors \( \mathbf{v}_1, \ldots, \mathbf{v}_5 \).
**Vectors:**
The vectors given are:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -2 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_3 = \begin{bmatrix} 2 \\ -2 \\ -8 \\ 0 \end{bmatrix}, \quad
\mathbf{v}_4 = \begin{bmatrix} 2 \\ -1 \\ 10 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_5 = \begin{bmatrix} 3 \\ -1 \\ -6 \\ 9 \end{bmatrix}
\]
**Task Explanation:**
The task is to determine a set of vectors from the given list that form a basis for the vector space they span. This involves finding the span of these vectors and determining which are linearly independent to serve as a basis. The basis should have the smallest number of vectors that still span the space.
**Procedure:**
1. Assemble the vectors into a matrix.
2. Use techniques such as row reduction to determine the rank and identify a set of linearly independent vectors.
3. Extract those vectors that represent the basis of the vector space.
Understanding and performing each of these steps is essential for identifying the basis of the space effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b7dd8f1-dbab-4607-8b33-bde9974f18c9%2Fa39596dd-d885-453a-8d3f-a55c188a9b25%2Fad37m2q_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 9: Linear Algebra - Finding a Basis**
**Objective:**
Find a basis for the space spanned by the given vectors \( \mathbf{v}_1, \ldots, \mathbf{v}_5 \).
**Vectors:**
The vectors given are:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -2 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_3 = \begin{bmatrix} 2 \\ -2 \\ -8 \\ 0 \end{bmatrix}, \quad
\mathbf{v}_4 = \begin{bmatrix} 2 \\ -1 \\ 10 \\ 3 \end{bmatrix}, \quad
\mathbf{v}_5 = \begin{bmatrix} 3 \\ -1 \\ -6 \\ 9 \end{bmatrix}
\]
**Task Explanation:**
The task is to determine a set of vectors from the given list that form a basis for the vector space they span. This involves finding the span of these vectors and determining which are linearly independent to serve as a basis. The basis should have the smallest number of vectors that still span the space.
**Procedure:**
1. Assemble the vectors into a matrix.
2. Use techniques such as row reduction to determine the rank and identify a set of linearly independent vectors.
3. Extract those vectors that represent the basis of the vector space.
Understanding and performing each of these steps is essential for identifying the basis of the space effectively.
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