Testing for Linear Independence In Exercises 41-48, determine whether the set of vectors in P, is linearly independent or linearly dependent. 47. S= {7 - 3x + 4r, 6 + 2r - ,1 - &r + 5}
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- ProofIn Exercises 65and 66, prove that the set of vectors is linearly independent and spans R3. B={(1,2,3),(3,2,1),(0,0,1)}ProofProve that if A is not square, then either the row vectors of A or the column vectors of A form a linearly dependent set.Showing Linear Dependence In Exercises 53-56, show that the set is linearly dependent by finding a nontrivial linear combination vectors in the set whose sum is the zero vector. Then express one of the vectors in the set as a linear combinations of the other vectors in the set. S={(1,2,3,4),(1,0,1,2),(1,4,5,6)}
- Proof When the set of vectors {u1,u2,...,un} is linearly independent and the set {u1,u2,...,un,v} is linearly dependent, prove that v is the linear combination of the uis.Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.Determining a Relationship Between Two Vectors In Exercises 47-54, determine whether u and v are orthogonal, parallel, or neither. u=(0,3,4), v=(1,8,6).
- True or False? In Exercises 79 and 80, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. a If dim(V)=n, then any set of n+1 vectors in V must be linearly dependent. b If dim(V)=n, then any set of n1 vectors in V must be linearly independent.Vector Operations In Exercises 2932, find a uv, b 2(u+3v), c 2vu. u=(7,0,0,0,9),v=(2,3,2,3,3)Proof Let A be an nn square matrix. Prove that the row vectors of A are linearly dependent if and only if the column vectors of A are linearly dependent.