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4 Listen From the list below, select all TRUE statements. (But do not select any false statements!) Of A is an m x n matrix with m pivots, then the matrix equation Ax = b is consistent for all b ERM If 2v1 + 3v₂ +4v3 SAMAN = 5v₁ + 7v2 + 9v3, then the vectors {V1, V2, V3} are linearly dependent. A transformation T: VW between two vector spaces is linear if and only if T(V1 + V₂) = T(v₁) +T(V₂) for all vectors V1, V2 E V.

4 Listen From the list below, select all TRUE statements. (But do not select any false statements!) Of A is an m x n matrix with m pivots, then the matrix equation Ax = b is consistent for all b ERM If 2v1 + 3v₂ +4v3 SAMAN = 5v₁ + 7v2 + 9v3, then the vectors {V1, V2, V3} are linearly dependent. A transformation T: VW between two vector spaces is linear if and only if T(V1 + V₂) = T(v₁) +T(V₂) for all vectors V1, V2 E 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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Listen From the list below, select all TRUE statements. (But do not select any false statements!) Olf A is an m x n matrix with m pivots, then the matrix equation Ax = b is consistent for all be Rm. Of 2v₁ +3v₂ +4v3 = 5v₁ + 7V2 + 9v3, then the vectors {V1, V2, V3} are linearly dependent. A transformation T: VW between two vector spaces is linear if and only if T(V1 + V₂) = T(v₁) +T(V₂) for all vectors V1, V2 EV. Of A and B are two invertible n x n matrices, then (AB)-¹ = A-¹B-¹ The determinant of a triangular matrix is the sum of the entries on the main diagonal. For any m X n matrix A, dim Nul(A) + dim Row(A) = n. If dim V = n, and S {V1, V2,..., Vn} is a set of n linearly independent vectors in V, then S is a basis for V. Eigenvalues of a matrix must be nonzero scalars. Eigenvectors of a matrix must be nonzero vectors. Switching two rows in a square matrix preserves its rank. Switching two rows in a square matrix preserves its characteristic polynomial. For an n x n matrix, the sum of the geometric multiplicities of all its eigenvalues always equals to n.
Transcribed Image Text:Listen From the list below, select all TRUE statements. (But do not select any false statements!) Olf A is an m x n matrix with m pivots, then the matrix equation Ax = b is consistent for all be Rm. Of 2v₁ +3v₂ +4v3 = 5v₁ + 7V2 + 9v3, then the vectors {V1, V2, V3} are linearly dependent. A transformation T: VW between two vector spaces is linear if and only if T(V1 + V₂) = T(v₁) +T(V₂) for all vectors V1, V2 EV. Of A and B are two invertible n x n matrices, then (AB)-¹ = A-¹B-¹ The determinant of a triangular matrix is the sum of the entries on the main diagonal. For any m X n matrix A, dim Nul(A) + dim Row(A) = n. If dim V = n, and S {V1, V2,..., Vn} is a set of n linearly independent vectors in V, then S is a basis for V. Eigenvalues of a matrix must be nonzero scalars. Eigenvectors of a matrix must be nonzero vectors. Switching two rows in a square matrix preserves its rank. Switching two rows in a square matrix preserves its characteristic polynomial. For an n x n matrix, the sum of the geometric multiplicities of all its eigenvalues always equals to n.
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