Page 497 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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(b) Use your method to find an orthonormal basis for the plane .
Find vectors x and y in that are orthonormal with respect to the inner product
35. but are not orthonormal with respect to the Euclidean inner product.
If W is a line through the origin of with the Euclidean inner product, and if u is a vector in
36. , then Theorem 6.3.4 implies that u can be expressed uniquely as , where is a
vector in W and is a vector in . Draw a picture that illustrates this.
Indicate whether each statement is always true or sometimes false. Justify your answer by
37. giving a logical argument or a counterexample.
(a) A linearly dependent set of vectors in an inner product space cannot be orthonormal.
(b) Every finite-dimensional vector space has an orthonormal basis.
(c) is orthogonal to in any inner product space.
(d) Every matrix with a nonzero determinant has a -decomposition.
What happens if you apply the Gram–Schmidt process to a linearly dependent set of vectors?
38.
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