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Use the method of Example 3 to show that the following set of vectors forms a basis for R³. {(3, 1, -4), (2, 5, 6), (1, 4, 8)}

Use the method of Example 3 to show that the following set of vectors forms a basis for R³. {(3, 1, -4), (2, 5, 6), (1, 4, 8)}

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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I need help with question 2 please? It has to do with coordinates and basis. The picture attatched to this question is an example that has to do with the direction of the question.

2. Use the method of Example 3 to show that the following set of vectors forms a basis for R³. {(3, 1, –4), (2, 5, 6), (1, 4, 8)}
Transcribed Image Text:2. Use the method of Example 3 to show that the following set of vectors forms a basis for R³. {(3, 1, –4), (2, 5, 6), (1, 4, 8)}
► EXAMPLE 3 Another Basis for A Show that the vectors v = (1, 2, 1), v= (2,9,0), and v ,= (3, 3,4) form a basis for R '. Solution We must show that these vectors are linearly independent and span R. To prove linear independence we must show that the vector equation CIv+ cv2+ C3V3 = 0 (1) has only the trivial solution; and to prove that the vectors span R' we must show that every vector b= (b1, b2, b3) in R' can be expressed as n+ C>V2 + C;V3 = b (2) By equating corresponding components on the two sides, these two equations can be expressed as the linear systems a+ 2c, + 3c = 0 a+ 2c2 + 3c = bị 2c1 +92 +3es= 0 and 2e1 +9c2 + 3c a = bị (3) +4c, = 0 +4c= by (verify). Thus, we have reduced the problem to showing that in (3) the homogeneous system has only the trivial solution and that the nonhomogeneous system is consistent for all values ofb.b, and by But the two systems have the same coefficient matrix 2 3 29 104 A3= 13 so it follows from parts (b). (e), and (g) of Theorem 2.3.8 that we can prove both results at the same time by showing that det(AD=0We leave it for you to confirm that det (A) - V. V and , torm a basis for A 1. which proves that the vectors
Transcribed Image Text:► EXAMPLE 3 Another Basis for A Show that the vectors v = (1, 2, 1), v= (2,9,0), and v ,= (3, 3,4) form a basis for R '. Solution We must show that these vectors are linearly independent and span R. To prove linear independence we must show that the vector equation CIv+ cv2+ C3V3 = 0 (1) has only the trivial solution; and to prove that the vectors span R' we must show that every vector b= (b1, b2, b3) in R' can be expressed as n+ C>V2 + C;V3 = b (2) By equating corresponding components on the two sides, these two equations can be expressed as the linear systems a+ 2c, + 3c = 0 a+ 2c2 + 3c = bị 2c1 +92 +3es= 0 and 2e1 +9c2 + 3c a = bị (3) +4c, = 0 +4c= by (verify). Thus, we have reduced the problem to showing that in (3) the homogeneous system has only the trivial solution and that the nonhomogeneous system is consistent for all values ofb.b, and by But the two systems have the same coefficient matrix 2 3 29 104 A3= 13 so it follows from parts (b). (e), and (g) of Theorem 2.3.8 that we can prove both results at the same time by showing that det(AD=0We leave it for you to confirm that det (A) - V. V and , torm a basis for A 1. which proves that the vectors
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