Show that {u,, u2, uz) is an orthogonal basis for R. Then express x as a linear combination of the u's. 3 2 1 4 u, = - 3 u, = 2 u3 = and x= - 3 - 1 4 1 Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of R"? Select all that apply. O A. The vectors must all have a length of 1. O B. The vectors must span W. O c. The vectors must form an orthogonal set. O D. The distance between any pair of distinct vectors must be constant. Which theorem could help prove one of these criteria from another? O A. If S= {u,, .., u, is an orthogonal set of nonzero vectors in R", then S is linearly independent and hence is a basis for the subspace spanned by S. O B. If S = {u,, ., u, and each u; has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S. Oc. If S= {u, ., u,} and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set. O D. If S= {u,, ..., u, is a basis in RP, then the members of S span RP and hence form an orthogonal set. Which calculations should be performed next? (Simplify your answers.) O A. u, •u1 = O B. u1 - u2 = OC. u •u2 = u, - u3 = u2 - u3 = u2 • u2 = u, *uz = u3 •u3 = u2 • u3 = DOC

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.