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1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³

1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 22EQ
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1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ = (1,1,1), V₂ = (1,1,0) and V3 = (1,0,0) 7- Find the coordinates of the vector U = (2√2,2,0) with respect to the orthonormal basis V₁ = (1, 0, 1), V₂ = 0, and V3 = (0,1,0) 8- If u = (1, 2, 3) and v = (2,0,1), The Inner product of the two vectors is <u,v> 9- The cosine of the angle between u = (1,1,-1) and v= : (2,1,1) is given by 10- The norm of the vector u = (3,3,1) is given by 11- Find Echelon form for the following matrix a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A [1 12- 3- Find Echelon form for the following matrix A=2 L4 a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A A = [2 13- Find Echelon form for the following matrix A= 1 L4 a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A 1 -1 1 2 3 22 2 3 5 1 21 −1 1 1 5. 1 3 5 7 8.
Transcribed Image Text:1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ = (1,1,1), V₂ = (1,1,0) and V3 = (1,0,0) 7- Find the coordinates of the vector U = (2√2,2,0) with respect to the orthonormal basis V₁ = (1, 0, 1), V₂ = 0, and V3 = (0,1,0) 8- If u = (1, 2, 3) and v = (2,0,1), The Inner product of the two vectors is <u,v> 9- The cosine of the angle between u = (1,1,-1) and v= : (2,1,1) is given by 10- The norm of the vector u = (3,3,1) is given by 11- Find Echelon form for the following matrix a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A [1 12- 3- Find Echelon form for the following matrix A=2 L4 a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A A = [2 13- Find Echelon form for the following matrix A= 1 L4 a. find the basis of the row space of A b. find the basis of the column space of A c. find the basis of the null space of A 1 -1 1 2 3 22 2 3 5 1 21 −1 1 1 5. 1 3 5 7 8.
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