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Mathematics, 24.04.2020 18:24 gg808
Recall that if we have m vectors u1, U2, ..., um in R", then we can form the matrix A whose columns are u1, ... , Um. Let B be the echelon form of A. Most questions have a yes no answer, but I am mostly interested in your reasons for the answer. Give full reasons for all answers. Suppose we are given the following matrix B: (3 0 -1 5 0 0 2 -1 10000)
(a) What is n?
(b) What is m?
(c) Are ui, ..., Um linearly independent?
(d) Does {U1, ... , um} span R ?
(e) Looking at B can you write down a subset of the original set {U1, ..., Um} that would be guaranteed to be linearly independent?
(f) Is there a subset of the original set {u1, ... , Um} that would be guaranteed to span R"?
(g) Write down a b e Rn for which Bx = b does not have a solution.
(h) Write down a b e R" for which Bx = b has a unique solution.
(i) Is there a new vector w E Rthat you could add to the set {u1, ..., Um} to guarantee that {U1, ... , um, w} will span R ?
(i) Is there a column of B that is in the span of the rest? If so, find it (k) Put B into reduced echelon form.
Write down a non-zero solution of Ax = 0 if you can. (m) How many free variables are there in the set of solutions to Ax = b when there is a solution? (n) If you erased the last row of zeros in B then would the columns of the resulting matrix be linearly independent?
Can you add rows to B to make the columns of the new matrix linearly independent? If yes, give an example of the new matrix you would construct.
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Answers: 3
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Recall that if we have m vectors u1, U2, ..., um in R", then we can form the matrix A whose columns...
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