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8. Prove or disprove the following: (a) The yz-plane is a subspace of R³. (b) The set of n x n upper triangular matrices is a subspace of M₂ (R). (c) The nxn invertible matrices is a subspace of M₂ (R). (d) If V is a vector space, W₁ and W₂ are subspaces of V, then i. W₁ W₂ is a subspace of V. ii. W₁ UW₂ is a subspace of V. iii. If W₁ W₂0, then W₁/W₂2 is a subspace of V.

8. Prove or disprove the following: (a) The yz-plane is a subspace of R³. (b) The set of n x n upper triangular matrices is a subspace of M₂ (R). (c) The nxn invertible matrices is a subspace of M₂ (R). (d) If V is a vector space, W₁ and W₂ are subspaces of V, then i. W₁ W₂ is a subspace of V. ii. W₁ UW₂ is a subspace of V. iii. If W₁ W₂0, then W₁/W₂2 is a subspace of V.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 51EQ
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### Linear Algebra: Subspaces In this exercise, we explore various examples to determine whether they are subspaces of given vector spaces. #### Problem Statement: **8. Prove or disprove the following:** (a) The *yz*-plane is a subspace of \(\mathbb{R}^3\). (b) The set of \(n \times n\) upper triangular matrices is a subspace of \(M_n(\mathbb{R})\). (c) The \(n \times n\) invertible matrices is a subspace of \(M_n(\mathbb{R})\). (d) If \(V\) is a vector space, \(W_1\) and \(W_2\) are subspaces of \(V\), then i. \(W_1 \cap W_2\) is a subspace of \(V\). ii. \(W_1 \cup W_2\) is a subspace of \(V\). iii. If \(W_1 \cap W_2 \neq \emptyset\), then \(W_1/W_2\) is a subspace of \(V\). ### Detailed Explanation: **(a) The *yz*-plane as a subspace of \(\mathbb{R}^3\):** The *yz*-plane consists of all points in \(\mathbb{R}^3\) where the *x*-coordinate is zero. In more formal terms, it includes all vectors of the form \((0, y, z)\). To verify if this plane is a subspace, we must check: - **Closure under addition:** If we add any two vectors from the *yz*-plane, the result must still be in the *yz*-plane. - **Closure under scalar multiplication:** If we multiply any vector in the *yz*-plane by a scalar, the resultant vector must still be in the *yz*-plane. **(b) The set of \(n \times n\) upper triangular matrices as a subspace of \(M_n(\mathbb{R})\):** Upper triangular matrices are matrices where all entries below the main diagonal are zero. This set will be a subspace of \(M_n(\mathbb{R})\) (the space of all \(n \times n\) matrices with real entries) if: - **Closure
Transcribed Image Text:### Linear Algebra: Subspaces In this exercise, we explore various examples to determine whether they are subspaces of given vector spaces. #### Problem Statement: **8. Prove or disprove the following:** (a) The *yz*-plane is a subspace of \(\mathbb{R}^3\). (b) The set of \(n \times n\) upper triangular matrices is a subspace of \(M_n(\mathbb{R})\). (c) The \(n \times n\) invertible matrices is a subspace of \(M_n(\mathbb{R})\). (d) If \(V\) is a vector space, \(W_1\) and \(W_2\) are subspaces of \(V\), then i. \(W_1 \cap W_2\) is a subspace of \(V\). ii. \(W_1 \cup W_2\) is a subspace of \(V\). iii. If \(W_1 \cap W_2 \neq \emptyset\), then \(W_1/W_2\) is a subspace of \(V\). ### Detailed Explanation: **(a) The *yz*-plane as a subspace of \(\mathbb{R}^3\):** The *yz*-plane consists of all points in \(\mathbb{R}^3\) where the *x*-coordinate is zero. In more formal terms, it includes all vectors of the form \((0, y, z)\). To verify if this plane is a subspace, we must check: - **Closure under addition:** If we add any two vectors from the *yz*-plane, the result must still be in the *yz*-plane. - **Closure under scalar multiplication:** If we multiply any vector in the *yz*-plane by a scalar, the resultant vector must still be in the *yz*-plane. **(b) The set of \(n \times n\) upper triangular matrices as a subspace of \(M_n(\mathbb{R})\):** Upper triangular matrices are matrices where all entries below the main diagonal are zero. This set will be a subspace of \(M_n(\mathbb{R})\) (the space of all \(n \times n\) matrices with real entries) if: - **Closure
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