GATE EC 2020 Set 1 — Question 11
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Engineering Mathematics → Linear Algebra → Linear Independence
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Question
If are six vectors in , which one of the following statements is FALSE?
Correct answer
(C) Any four of these vectors form a basis for R⁴.
Solution
In , the dimension of the space is 4.
(A) True: A set of vectors spans a space only if it contains at least as many linearly independent vectors as the dimension of the space. Six vectors might be linearly dependent and fail to span .
(B) True: In any -dimensional space, any set of more than vectors is linearly dependent. Since , these vectors must be linearly dependent.
(C) False: For a set of 4 vectors to form a basis for , they must be linearly independent. Not any arbitrary subset of four vectors from the given six will necessarily be linearly independent.
(D) True: In an -dimensional space, any set of vectors that spans the space is automatically a basis.
(A) True: A set of vectors spans a space only if it contains at least as many linearly independent vectors as the dimension of the space. Six vectors might be linearly dependent and fail to span .
(B) True: In any -dimensional space, any set of more than vectors is linearly dependent. Since , these vectors must be linearly dependent.
(C) False: For a set of 4 vectors to form a basis for , they must be linearly independent. Not any arbitrary subset of four vectors from the given six will necessarily be linearly independent.
(D) True: In an -dimensional space, any set of vectors that spans the space is automatically a basis.
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