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you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R² that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m

you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false. (a) The null space of an m x n matrix A with real elements is a subspace of Rm. (b) The solution set of any linear system of m equations in n variables forms a subspace of C". (c) The points in R² that lie on the line y = mx + b form a subspace of R2 if and only if b = 0. (d) If m

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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# Subspaces and Problems ## True or False Questions For Questions (a)–(d), state if the given statement is true or false, and provide justification. If true, you can quote a definition or theorem. If false, provide an example, illustration, or explanation. **(a)** The null space of an \(m \times n\) matrix \(A\) with real elements is a subspace of \(\mathbb{R}^m\). **(b)** The solution set of any linear system of \(m\) equations in \(n\) variables forms a subspace of \(\mathbb{C}^n\). **(c)** The points in \(\mathbb{R}^2\) that lie on the line \(y = mx + b\) form a subspace of \(\mathbb{R}^2\) if and only if \(b = 0\). **(d)** If \(m < n\), then \(\mathbb{R}^m\) is a subspace of \(\mathbb{R}^n\). ## Problems ### Problem 1 Let \(S = \{ x \in \mathbb{R}^3 : x = (r - 2s, 3r + s, s), \, r, s \in \mathbb{R} \}\). **(a)** Show that \(S\) is a subspace of \(\mathbb{R}^3\). **(b)** Show that the vectors in \(S\) lie on the plane with equation \(3x - y + 7z = 0\). ### Problem 2 Let \(S = \{ x \in \mathbb{R}^2 : x = (2k, -3k), \, k \in \mathbb{R} \}\). **(a)** Show that \(S\) is a subspace of \(\mathbb{R}^2\). **(b)** Make a sketch depicting the subspace \(S\) in the Cartesian plane. ## Set Notation and Subspaces For Problems 3–22, express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). ### Problem 3 \(V = \mathbb{R}^
Transcribed Image Text:# Subspaces and Problems ## True or False Questions For Questions (a)–(d), state if the given statement is true or false, and provide justification. If true, you can quote a definition or theorem. If false, provide an example, illustration, or explanation. **(a)** The null space of an \(m \times n\) matrix \(A\) with real elements is a subspace of \(\mathbb{R}^m\). **(b)** The solution set of any linear system of \(m\) equations in \(n\) variables forms a subspace of \(\mathbb{C}^n\). **(c)** The points in \(\mathbb{R}^2\) that lie on the line \(y = mx + b\) form a subspace of \(\mathbb{R}^2\) if and only if \(b = 0\). **(d)** If \(m < n\), then \(\mathbb{R}^m\) is a subspace of \(\mathbb{R}^n\). ## Problems ### Problem 1 Let \(S = \{ x \in \mathbb{R}^3 : x = (r - 2s, 3r + s, s), \, r, s \in \mathbb{R} \}\). **(a)** Show that \(S\) is a subspace of \(\mathbb{R}^3\). **(b)** Show that the vectors in \(S\) lie on the plane with equation \(3x - y + 7z = 0\). ### Problem 2 Let \(S = \{ x \in \mathbb{R}^2 : x = (2k, -3k), \, k \in \mathbb{R} \}\). **(a)** Show that \(S\) is a subspace of \(\mathbb{R}^2\). **(b)** Make a sketch depicting the subspace \(S\) in the Cartesian plane. ## Set Notation and Subspaces For Problems 3–22, express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). ### Problem 3 \(V = \mathbb{R}^
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