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proof
If W is a 3-dimensional subspace of an 6-dimensional
V , there is a 2-dimensional subspace U of V so that U∩W = {0}.
Step by step
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- Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.Proof Let A be a fixed mn matrix. Prove that the set W={xRn:Ax=0} is a subspace of Rn.Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose second component is the square of the first.
- Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are nonnegative.Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are integers.Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are rational numbers.
- Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are Pythagorean triples.Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose first component is 2.Proof Let V and W be two subspaces of vector space U. (a) Prove that the set V+W={u:u=v+w,vVandwW} is a subspace of U. (b) Describe V+W when V and W are subspaces of U=R2: V={(x,0):xisarealnumber} and W={(0,y):yisarealnumber}.
- True or False?In Exercises 43 and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If W is a subspace of a vector space V, then it has closure under scalar multiplication as defined in V. (b) If V and W are both subspaces of vector space U, then the intersection of V and W is also a subspace. (c) If U, V, and W are vector spaces such that W is a subspace of V and U is a subspace of V, then W=U.Verifying Subspaces In Exercises 1-6, verify that W is a subspace of V. In each case assume that V has the standard operations. Calculus W is the set of all functions that are continuous on [1,1]. V is the set of all functions that are integrable on [1,1].Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all non-negative functions in C(,).