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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.2: Inner Product Spaces

Problem 94E

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Define spanning set. Determine whether the vectors u,
(1,2,0),
Uz = (-1,1,2) and uz
= (3,0, –4) span R³.

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Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)
For which values of t is each set linearly independent? a S={(t,1,1),(1,t,1),(1,1,t)} b S={(t,1,1),(1,0,1),(1,1,3t)}
Define spanning set. Determine whether the vectors u1 = (1,2,1), u2 = (−1,1,2) andu3 = (2,0, −4) span R3.
Consider the vectors u1 = (1, 2, 1), u2 = (0, 3, 2) and u3 = (−2, 1, 1). Determine whether the set S = {u1, u2, u3} is a spanning set for R3.
Let {x1, x2,...,xk} be a spanning set for a vector space V. (a) If we add another vector, xk+1, to the set, will we still have a spanning set? Explain. (b) If we delete one of the vectors, say, xk, from the set, will we still have a spanning set? Explain.
Suppose {u1, ...., ur, w1, ...., ws } is a linearly independent subset of V. Show that Span {ui} ∩ Span{wj}={0}.
Which of the following sets is a spanning set, but not a basis for the space V={(a,b,c) ∈ R^3 | a+b=0}?
Determine whether the set S=(U,V,W) are linearly independent or linearly dependent. Where, U= (1,-3,-2), V= (-1,2,1) and W= (-3,5,2)
Determine if S is a spanning set for Rn & if Sis linearly independent. Determine if the given vector ~v is in the span of S if YES, determine if the given vector ~ can be written as a linear combination of the S vectors.
Write the following sets as a span of the minimum number of vectors. If this is not a span and why?
Determine if S is a spanning set for Rn & if S is linearly independent. Determine if the given vector ~v is in the span of S if YES, determine if the given vector ~v can be written as a linear combination of the S vectors.
Determine whether the set S is linearly independent or linearly dependent. S = {(7, 0, 0), (0, 9, 0), (0, 0, −16), (7, 5,−8)}

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