Determine if the statements are true or false. 1. Any four vectors in R³ are linearly dependent. 2 2. Any four vectors in R³ span R³ ? V V 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). 2 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.) ? 6. An n x n matrix A is diagonalizable 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- ? V PEW ? V
Determine if the statements are true or false. 1. Any four vectors in R³ are linearly dependent. 2 2. Any four vectors in R³ span R³ ? V V 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). 2 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.) ? 6. An n x n matrix A is diagonalizable 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- ? V PEW ? V
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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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 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.) ?
6. An n x n matrix A is diagonalizable if and only if it has a distinct eigenvalues. ?
7. Let W be a subspace of R". If p is the projection of b onto W, then b-peW¹ ?
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