In Exercises 1 and 2, determine if the vectors are linearly independent. Justify
ID: 3138408 • Letter: I
Question
In Exercises 1 and 2, determine if the vectors are linearly independent. Justify each answer 1· 101. 1 -30 3. Determine if the columns of the matrix form a linearly independent set. Justify your answer. 1 0 3 4. (a) For what values of h is va in Span i, v2), and (b) for what values of h is {vi, v2, v3j linearly dependent? Justify your answer 2 6 In Exercises 5, 6, and 7, determine by inspection whether the give sets of vectors are linear independent Justify your answer. 5.-2 6 8. How many pivot columns must a 5 x 7 matrix have fts columns span R5? Why? 9. a. Fill in the blank in the following statement: "If A is an m x n matrix, then the columns of A are pivot columns." linearly independent if and only if A has b. Explain why the statement in (a) is true 10. Suppose an m × 72 matrix A has n pivot columns. Explain why for each b in Rrn the equation Ax-b has at most one solution. (Hint: Explain why Ax - b cannot have infinitely many solutions.)Explanation / Answer
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The RREF of A is I3 which implies that the given vectors are linearly independent.
2. Since (-2,-8)T is not a scalar multiple of (-1,4)T, hence, the given vectors are linearly independent.
3. Let the given matrix be denoted by A. The RREF of A is
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It implies that the columns of A are linearly independent.
4. It may be observed that v2 = (-2,10,6)T = -2(1,-5,-3)T = -2v1. Hence span{v1,v2} = span{v1}. Now, v3 = (2,-9,hT) will be in span{v1,v2} = span{v1} if v3 is a scalar multiple of v1. Let v3 = kv1, where k is an arbitrary scalar. Then, 2 = k*1, -9 = k*(-5) and h = k*(-3), i.e. k = 2, k = 9/5 and h = -3k. Apparently, this is not possible as k cannot have 2 values. Hence , v3 is not a scalar multiple of v1 so that, regardless of the value of h, v3 cannot be in span{v1,v2} = span{v1}. Also, the set { v1,v2, v3} will always be linearly dependent as v2 =-2v1.
Please post the remaining questions again.
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