Vector Spaces : Show that the following set S of vectors is linearly independent in V4(R)

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  Show that if {v1, v2, v3} is linearly dependent set of vectors in a vector space V, and v4 is any vector in V, then {v1, v2, v3, v4} is also linear dependent. Please see the attached file.

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  Select a maximal set of linearly independent vectors in S. Every vector not in this set is a linear combination of those vectors.

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  Therefore, u and v are also in W. A basis for a vector space is any set of lineraly independent vectors which span the whole vector space W.

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  Since if and only if x=0, i.e., all components of x are 0's, then by definition, are linearly independent vectors. This also means that the matrix A is of full rank, or A is invertible, and therefore the inverse exists, which is denoted as .

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  Now we can use these vectors to write a new one by linearly adding them together: (1.1) These new vectors form a vector space W (1.2) and the set S is called the "spanning set" of W.