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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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.2: Length And Angle: The Dot Product
Problem 17EQ
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Determine if the statements are true or false. 1. Any four vectors In R³ are linearly dependent. True 2. Any four vectors in R³ span R³. True 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 V 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 6. An n x n matrix A is diagonalizable if and only if it has n distinct eigenvalues. False v 7. Let W be a subspace of R". If p is the projection of b onto W, then b-p & W True
Transcribed Image Text:Determine if the statements are true or false. 1. Any four vectors In R³ are linearly dependent. True 2. Any four vectors in R³ span R³. True 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 V 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 6. An n x n matrix A is diagonalizable if and only if it has n distinct eigenvalues. False v 7. Let W be a subspace of R". If p is the projection of b onto W, then b-p & W True
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