Find the Null Space [[1,0,0,1],[0,1,0,-3],[0,0,1,1]]

Problem

Null Space*[[1,0,0,1],[0,1,0,−3],[0,0,1,1]]

Solution

1. Identify the matrix A and the equation for the null space A*x=0 The given matrix is already in reduced row echelon form (RREF).

A=[[1,0,0,1],[0,1,0,−3],[0,0,1,1]]

2. Determine the pivot columns and free variables. Columns 1, 2, and 3 contain pivots. Column 4 does not contain a pivot, so (x_4) is a free variable.

(x_4)=t

3. Write the system of equations corresponding to the rows of the matrix to express the pivot variables in terms of the free variable.

(x_1)+(x_4)=0⇒(x_1)=−(x_4)

(x_2)−3*(x_4)=0⇒(x_2)=3*(x_4)

(x_3)+(x_4)=0⇒(x_3)=−(x_4)

4. Express the general solution vector x in terms of the parameter t

x=[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[−t],[3*t],[−t],[t]]=t*[[−1],[3],[−1],[1]]

5. Define the null space as the span of the resulting vector.

Null(A)=span*{[−1],[3],[−1],[1]}

Final Answer

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

Want more problems? Check here!