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Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)

Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)

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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Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)
Transcribed Image Text:Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)
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