410 Ch. 5 Theory of Systems of Linear Equations and Elgeavalue/Elgnvector Proble

Question

410 Ch. 5 Theory of Systems of Linear Equations and Elgeavalue/Elgnvector Problem (8) The coefficient matrix X in the six regression equations in Exam- ple 6. 6. For the following matrices, describe the set of vectors in the null space. -3 4 1 T2 1 5 01 2 11 (b) 1 2 4 3 [113-1 1 1 0 0 11 (c) 0 0 1 1 0 0 1 0 1 o 7. (a) For A in Exercise 6, part (a) and b-s [ 10, 101, if Ax b has the given solution xI0, 0, 10], find the family of all solutions to (b) Find a solution to Ax b in part (a) with x, 3. 8. (a) For A in Exercise 6, part (b) and b 130, 30, 20], if Ax b has the given solution x' [10, 10, 0, 0], find the family of all solutions to Ax b. (b) Find a solution to Ax-b in part (a) with x, 5. 9. (a) For A in Exercise 6, part (c) and b 110, 15, 5, 0, 15]. if Ax b has the given solution x' -15, 0, 5, 0, 5], find the family of all solutions to Ax = b (b) Find a solution to Ax b in part (a) with 10 and 10. (c) Find a solution to Ax-b in part (a) with x, 10 and x2 = 5. 10. Consider the modified refinery system from Example 1:

Explanation / Answer

Reduce matrix A to row echelon form:

Add (-1/2 * row1) to row2

Add (-1/2 * row1) to row3

Add (-1/3 * row2) to row3

First, we must reduce the matrix so we can calculate the pivots of the matrix (note that we are reducing to row echelon form, not reduced row echelon form):

The matrix has 2 pivots (hilighted above in yellow)
Because we have found pivots in columns 0 and 1. We know that these columns in the original matrix define the Column Space(i.e range vector) of the matrix.
Therefore, the Column Space(i.e range vector) is given by the following equation:

b)

no info about theorm (i)

2 1 5 0 0 3/2 3/2 -3 1 1 3 -1

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