Find the Null Space [[1,-2,-4,-5,0],[0,1,2,1,0]]

Problem

Null Space*[[1,−2,−4,−5,0],[0,1,2,1,0]]

Solution

1. Identify the matrix A and the system of equations A*x=0 We are looking for all vectors x=[(x_1),(x_2),(x_3),(x_4),(x_5)]T such that:

[[1,−2,−4,−5,0],[0,1,2,1,0]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[0],[0]]

2. Perform row reduction to find the Reduced Row Echelon Form (RREF). Add 2 times the second row to the first row:

(R_1)+2*(R_2)→(R_1)

[[1,0,0,−3,0],[0,1,2,1,0]]

3. Write the equations corresponding to the RREF matrix:

(x_1)−3*(x_4)=0

(x_2)+2*(x_3)+(x_4)=0

4. Identify free variables by looking for columns without pivots. The pivots are in columns 1 and 2, so (x_3) (x_4) and (x_5) are free variables. Let (x_3)=s() (x_4)=t and (x_5)=u

5. Express pivot variables in terms of the free variables:

(x_1)=3*t

(x_2)=−2*s−t

6. Write the general solution in vector form:

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=[[3*t],[−2*s−t],[s()],[t],[u]]

[[(x_1)],[(x_2)],[(x_3)],[(x_4)],[(x_5)]]=s*[[0],[−2],[1],[0],[0]]+t*[[3],[−1],[0],[1],[0]]+u*[[0],[0],[0],[0],[1]]

Final Answer

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

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