Determine if the Vector is in the Column Space

Problem

A=[[2,−3,0],[1,4,−3]],b=[[7],[−6]]

Solution

1. Identify the goal, which is to determine if the vector b is in the column space of matrix A denoted Col *A This is true if the linear system A*x=b has at least one solution.

2. Set up the augmented matrix [A|b] to solve the system of linear equations.

[[2,−3,0,|,7],[1,4,−3,|,−6]]

3. Perform row operations to reach row echelon form. Swap Row 1 and Row 2.

[[1,4,−3,|,−6],[2,−3,0,|,7]]

4. Eliminate the first entry in Row 2 by replacing Row 2 with (R_2)−2*(R_1)

[[1,4,−3,|,−6],[0,−11,6,|,19]]

5. Analyze the consistency of the system. The augmented matrix is in row echelon form and does not contain a row of the form [0,0,0|k] where k≠0

6. Conclude that because there is no pivot in the augmented column, the system is consistent. In fact, since there are more variables than pivots, there are infinitely many solutions. Since a solution exists, the vector is in the column space.

Final Answer

b∈Col *A* is True

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