Find the Null Space [[1,2,3,4],[0,1,2,3],[0,0,0,1]]

Problem

Null Space*[[1,2,3,4],[0,1,2,3],[0,0,0,1]]

Solution

1. Identify the matrix A and set up the homogeneous equation A*x=0 to find the vectors x in the null space.

A=[[1,2,3,4],[0,1,2,3],[0,0,0,1]]

2. Perform row reduction to find the Reduced Row Echelon Form (RREF). Subtract 3 times the third row from the second row, and 4 times the third row from the first row.

[[1,2,3,0],[0,1,2,0],[0,0,0,1]]

3. Continue row reduction by subtracting 2 times the second row from the first row.

[[1,0,−1,0],[0,1,2,0],[0,0,0,1]]

4. Identify the pivot columns and free variables. Columns 1, 2, and 4 have pivots. Column 3 corresponds to the free variable (x_3)

(x_1)−(x_3)=0

(x_2)+2*(x_3)=0

(x_4)=0

5. Express the variables in terms of the free variable (x_3)

(x_1)=(x_3)

(x_2)=−2*(x_3)

(x_4)=0

6. Write the solution in vector form by factoring out the free variable (x_3)

[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[(x_3)],[−2*(x_3)],[(x_3)],[0]]=(x_3)*[[1],[−2],[1],[0]]

Final Answer

Null Space*[[1,2,3,4],[0,1,2,3],[0,0,0,1]]=span*{[1],[−2],[1],[0]}

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