For each of the following, select the correct multiple choice response. (5 point
ID: 3036603 • Letter: F
Question
For each of the following, select the correct multiple choice response.
(5 points) Let V be a vector space, and let X be a subset of V. What does it mean when we say that X is linearly independent? 8. (a) All the elements of X are distinct from each other. (b) X is closed under both addition and scalar multiplication. (c) X is a basis of V. (d) The number of elements of X is less than or equal to the dimension of V. (e) Every element in V is a linear combination of elements in X. (f) The only way to write 0 as a near combination of elements of X is the zero combination (where one takes zero multiples of each element of X).Explanation / Answer
The answer is (f). When we say that X is linearly independent, the only way to write 0 as a linear combination of elements of X is the zero combination. The answer is (d). When we say that X is linearly dependent, there is a way to write 0 as a linear combination of elements of X other than the zero combination. The options (a), (d) and ( e) are all correct. Option ( c) is closest to the correct answer. A msquare matrix A is diagonalizable if it has distinct linearly independent eigenvectors, which in turn, depends on the entries of A. Option (d) is the correct answer. Option (f) is the correct answer.
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