Page 382 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) The set of matrices that contain exactly two 1's and two 0's is a linearly independent
set in .

(b) If is a linearly dependent set, then each vector is a scalar multiple of the other.

(c) If is a linearly independent set, then so is the set for every

nonzero scalar k.

(d) The converse of Theorem 5.3.2a is also true.

Show that if is a linearly dependent set with nonzero vectors, then each vector in the

25. set is expressible as a linear combination of the other two.

Theorem 5.3.3 implies that four nonzero vectors in must be linearly dependent. Give an
26. informal geometric argument to explain this result.

27.
(a) In Example 3 we showed that the mutually orthogonal vectors , , and form a linearly
independent set of vectors in . Do you think that every set of three nonzero mutually
orthogonal vectors in is linearly independent? Justify your conclusion with a geometric
argument.

(b) Justify your conclusion with an algebraic argument.

Hint Use dot products.

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