A basis of a vector space V is a maximal linearly independent set of vectors in...

A basis of a vector space V is a maximal linearly independent set of vectors in V . Similarly, one can view it as a minimal spanning set of vectors in V . Prove that any set S ⊆ V spanning a finite-dimensional vector space V contains a basis of V .

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Colby Messinger answered 2 years ago

Define a subspace of a vector space V . Take the set of vectors in Rn...

Define a subspace of a vector space V . Take the set of vectors in Rn such that th coordinates add up to 0. I that a subspace. What about the set whose coordinates add up to 1. Explain your answers.

If {v1, v2, v3, v4} is a linearly-independent subset of a vector space V over the...

If {v1, v2, v3, v4} is a linearly-independent subset of a vector space V over the field Q, is the set {3v1 + 2v2 + v3 + v4, 2v1 + 5v2, 3v3 + 2v4, 3v1 + 4v2 + 2v3 + 3v4} linearly independent as well?

Determine whether the members of the given set of vectors are linearly independent. If they are...

Determine whether the members of the given set of vectors are linearly independent. If they are linearly dependent, find a linear relation among them of the form c1x(1) + c2x(2) + c3x(3) = 0. (Give c1, c2, and c3 as real numbers. If the vectors are linearly independent, enter INDEPENDENT.) x(1) = 9 1 0 , x(2) = 0 1 0 , x(3) = −1 9 0

Determine whether each set of vectors is linearly dependent or linearly independent. a) (1,1,0,1), (1,0,1,1), (0,1,1,1)...

Determine whether each set of vectors is linearly dependent or linearly independent. a) (1,1,0,1), (1,0,1,1), (0,1,1,1) b) (1,0,1,0), (0,1,0,1), (1,-1,1,-1), (1,-1,0,0)

Prove the follwing statements Suppose that S is a linearly independent set of vectors in the...

Prove the follwing statements Suppose that S is a linearly independent set of vectors in the vector space V and let w be a vector of V that is not in S. Then the set obtained from S by adding w to S is linearly independent in V. If U is a subspace of a vector space V and dim(U)=dim(V), then U=V.

1. Determine each of the following set of vectors is linearly independent or dependent. (a) S1...

1. Determine each of the following set of vectors is linearly independent or dependent. (a) S1 = {(1, 2, 3),(4, 5, 6),(6, 9, 12)}. (b) S2 = {(1, 2, 3, 4),(5, 6, 7, 8),(3, 2, 1, 0)}. (c) S3 = {(1, 2, 3, 4),(5, 6, 7, 8),(9, 10, 11, 12)}

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

For each family of vectors, determine wether the vectors are linearly independent or not, and in...

For each family of vectors, determine wether the vectors are linearly independent or not, and in case they are linearly dependent, find a linear relation between them. a) x1 = (2, 2, 0), x2 = (0, 2, 2), x3 = (1, 0, 1) b) x1 = (2, 1, 0), x2 = (0, 1, 0), x3 = (1, 2, 0) c) x1 = (1, 1, 0, 0), x2 = (0, 1, 1, 0), x3 = (0, 0, 1, 1), x4 =...

if {v1,v2,v3} is a linearly independent set of vectors, then {v1,v2,v3,v4} is too.

Determine if the column vectors [2,1,3,4]' [1,1,1,1]' and [5,3,7,9]' are linearly independent

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