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() (O (-- 1) The span in F of the vectors 2 is a plane. 2) If a, b is a basis of R2, there exists v€ R² such that both v, b and v, a are bases of R2. 3) The linear system I+z = 1 y +z = 1 I+y+2: = 2 has infinitely many solution.

() (O (-- 1) The span in F of the vectors 2 is a plane. 2) If a, b is a basis of R2, there exists v€ R² such that both v, b and v, a are bases of R2. 3) The linear system I+z = 1 y +z = 1 I+y+2: = 2 has infinitely many solution.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 48EQ
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O00 1) The span in F" of the vectors 2 6 is a plane. 2) If a, b is a basis of R², there exists v E R² such that both v, b and v, a are bases of R?. 3) The linear system I+z = 1 y +z = 1 x + y + 2z = 2 1 has infinitely many solution. 4) If the linear map T : C³ → C³ is diagonalizable and has 1,9 as its only eigenvalues eigenvalues, then its characteristic and minimal polynomials are the same. 5) If SE L(V) is an isometry on the inner product space V, then S- I is always invertible. 6) If N E L(V) is nilpotent then so is I+ N². 7) If the product AB of the matrices A € Matm,n(F) and B € Mat,,m(F) is non- singular so is the product BA. 8) (2 3 1 det 0 0 3 = 12. \0 2 3) 9) Consider R? with its Euclidean inner product. There exists three non-zero vectors in R², which are mutually orthogonal. 10) If A € Mat,, is diagonalizable it admits n linearly independent eigenvectors.
Transcribed Image Text:O00 1) The span in F" of the vectors 2 6 is a plane. 2) If a, b is a basis of R², there exists v E R² such that both v, b and v, a are bases of R?. 3) The linear system I+z = 1 y +z = 1 x + y + 2z = 2 1 has infinitely many solution. 4) If the linear map T : C³ → C³ is diagonalizable and has 1,9 as its only eigenvalues eigenvalues, then its characteristic and minimal polynomials are the same. 5) If SE L(V) is an isometry on the inner product space V, then S- I is always invertible. 6) If N E L(V) is nilpotent then so is I+ N². 7) If the product AB of the matrices A € Matm,n(F) and B € Mat,,m(F) is non- singular so is the product BA. 8) (2 3 1 det 0 0 3 = 12. \0 2 3) 9) Consider R? with its Euclidean inner product. There exists three non-zero vectors in R², which are mutually orthogonal. 10) If A € Mat,, is diagonalizable it admits n linearly independent eigenvectors.
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