Find the Kernel

Problem

[[1,1,1,1],[3,2,−1,2],[2,2,1,−1],[−1,3,2,−1]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

Solution

1. Define the kernel. The kernel (or null space) of a matrix A is the set of all vectors x such that A*x=0 We ignore the vector on the right side of the provided equation as finding a kernel always involves solving the homogeneous system.

2. Set up the augmented matrix. We represent the system A*x=0 using the coefficients of the matrix.

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

3. Perform row operations to reach row echelon form. Subtract 3 times row 1 from row 2, 2 times row 1 from row 3, and add row 1 to row 4.

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

4. Continue row reduction. Multiply row 2 by −1 Then add 4 times the new row 2 to row 4.

[[1,1,1,1,0],[0,1,4,1,0],[0,0,−1,−3,0],[0,0,−13,−4,0]]

5. Eliminate the last row. Multiply row 3 by −1 Subtract 13 times the new row 3 from row 4.

[[1,1,1,1,0],[0,1,4,1,0],[0,0,1,3,0],[0,0,0,35,0]]

6. Identify the pivot columns. Since there is a pivot in every column of the coefficient matrix, the only solution to the homogeneous system is the trivial solution.

(x_4)=0

(x_3)+3*(0)=0⇒(x_3)=0

(x_2)+4*(0)+0=0⇒(x_2)=0

(x_1)+0+0+0=0⇒(x_1)=0

Final Answer

Ker*([1,1,1,1],[3,2,−1,2],[2,2,1,−1],[−1,3,2,−1])={[0],[0],[0],[0]}

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