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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. ? V 3. The rank of a matrix is equal to the number of pivots in its RREF. ?

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. ? V 3. The rank of a matrix is equal to the number of pivots in its RREF. ?

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 R³ are linearly dependent. ? 2. Any four vectors in R³ span R³. ? 3. The rank of a matrix is equal to the number of pivots in Its RREF. ? 4. {V₁, V2,..., Vn} is a basis for span(V₁, V2,..., Vn). ? 5. If v Is an elgenvector 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.) ? 6. An n x n matrix A is dlagonalizable if and only if it has n distinct eigenvalues. ? 7. Let W be a subspace of R". If p is the projection of b onto W, then b-peW-. ? Note: You can earn partial credit on this problem. v V
Transcribed Image Text: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. ? 3. The rank of a matrix is equal to the number of pivots in Its RREF. ? 4. {V₁, V2,..., Vn} is a basis for span(V₁, V2,..., Vn). ? 5. If v Is an elgenvector 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.) ? 6. An n x n matrix A is dlagonalizable if and only if it has n distinct eigenvalues. ? 7. Let W be a subspace of R". If p is the projection of b onto W, then b-peW-. ? Note: You can earn partial credit on this problem. v V
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