Determine if the Vector is in the Column Space

Problem

A=[[1,2,−3],[0,1,2],[−1,2,0]],B=[[−1],[2],[0]],C=[[0],[4],[−3]]

Solution

1. Understand the definition of column space. A vector is in the column space of a matrix if it can be written as a linear combination of the columns of that matrix, which is equivalent to the system A*x=b being consistent.

2. Set up the augmented matrix for vector B to check if A*x=B has a solution.

[[1,2,−3,|,−1],[0,1,2,|,2],[−1,2,0,|,0]]

3. Perform row operations to reach row echelon form for B Add row 1 to row 3.

[[1,2,−3,|,−1],[0,1,2,|,2],[0,4,−3,|,−1]]

4. Continue row reduction by subtracting 4 times row 2 from row 3.

[[1,2,−3,|,−1],[0,1,2,|,2],[0,0,−11,|,−9]]

5. Conclude for vector B Since the system is consistent (there is a pivot in every row of the coefficient matrix or no row of the form [0,0,0|k] where k≠0, B is in the column space of A

6. Set up the augmented matrix for vector C to check if A*x=C has a solution.

[[1,2,−3,|,0],[0,1,2,|,4],[−1,2,0,|,−3]]

7. Perform row operations for C Add row 1 to row 3.

[[1,2,−3,|,0],[0,1,2,|,4],[0,4,−3,|,−3]]

8. Continue row reduction by subtracting 4 times row 2 from row 3.

[[1,2,−3,|,0],[0,1,2,|,4],[0,0,−11,|,−19]]

9. Conclude for vector C Since the system is consistent, C is in the column space of A

Final Answer

B∈Col(A),C∈Col(A)

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