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Determine if the statements are true or false. 1. Any four vectors In R³ are linearly dependent. 2. Any four vectors in R³ span R³. True V 3. The rank of a matrix is equal to the number of pivots in its RREF. False 4. {V1, V2,..., Vn} is a basis for span(V₁, V2,..., Vn). True 5. If v is an eigenvector of a matrix A, then v is an eigenvector of A+cI for all scalars c. (Here I denotes the identity matrix of the same dimension as A.) False True 6. An n x n matrix A is diagonalizable if and only if it has n distinct eigenvalues. False 7. Let W be a subspace of R". If p is the projection of b onto W, then b - p € W+. True V

Determine if the statements are true or false. 1. Any four vectors In R³ are linearly dependent. 2. Any four vectors in R³ span R³. True V 3. The rank of a matrix is equal to the number of pivots in its RREF. False 4. {V1, V2,..., Vn} is a basis for span(V₁, V2,..., Vn). True 5. If v is an eigenvector of a matrix A, then v is an eigenvector of A+cI for all scalars c. (Here I denotes the identity matrix of the same dimension as A.) False True 6. An n x n matrix A is diagonalizable if and only if it has n distinct eigenvalues. False 7. Let W be a subspace of R". If p is the projection of b onto W, then b - p € W+. True V

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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**Determine if the statements are true or false:** 1. Any four vectors in \(\mathbb{R}^3\) are linearly dependent. - **True** 2. Any four vectors in \(\mathbb{R}^3\) span \(\mathbb{R}^3\). - **True** 3. The rank of a matrix is equal to the number of pivots in its RREF. - **False** 4. \(\{v_1, v_2, \ldots , v_n\}\) is a basis for \(\text{span}(v_1, v_2, \ldots , v_n)\). - **True** 5. If \(\mathbf{v}\) is an eigenvector of a matrix \(A\), then \(\mathbf{v}\) is an eigenvector of \(A + cI\) for all scalars \(c\). Here, \(I\) denotes the identity matrix of the same dimension as \(A\). - **False** 6. An \(n \times n\) matrix \(A\) is diagonalizable if and only if it has \(n\) distinct eigenvalues. - **False** 7. Let \(W\) be a subspace of \(\mathbb{R}^n\). If \(p\) is the projection of \(\mathbf{b}\) onto \(W\), then \(\mathbf{b} - p \in W^{\perp}\). - **True**
Transcribed Image Text:**Determine if the statements are true or false:** 1. Any four vectors in \(\mathbb{R}^3\) are linearly dependent. - **True** 2. Any four vectors in \(\mathbb{R}^3\) span \(\mathbb{R}^3\). - **True** 3. The rank of a matrix is equal to the number of pivots in its RREF. - **False** 4. \(\{v_1, v_2, \ldots , v_n\}\) is a basis for \(\text{span}(v_1, v_2, \ldots , v_n)\). - **True** 5. If \(\mathbf{v}\) is an eigenvector of a matrix \(A\), then \(\mathbf{v}\) is an eigenvector of \(A + cI\) for all scalars \(c\). Here, \(I\) denotes the identity matrix of the same dimension as \(A\). - **False** 6. An \(n \times n\) matrix \(A\) is diagonalizable if and only if it has \(n\) distinct eigenvalues. - **False** 7. Let \(W\) be a subspace of \(\mathbb{R}^n\). If \(p\) is the projection of \(\mathbf{b}\) onto \(W\), then \(\mathbf{b} - p \in W^{\perp}\). - **True**
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