Find the Kernel

Problem

[[1,1,0,2,−3],[0,0,1,−5,4],[0,0,0,0,0]]*x=[[0],[0],[0]]

Solution

1. Identify the system of linear equations represented by the matrix A*x=0 The kernel (or null space) is the set of all vectors x=[(x_1),(x_2),(x_3),(x_4),(x_5)]T such that:

(x_1)+(x_2)+2*(x_4)−3*(x_5)=0

(x_3)−5*(x_4)+4*(x_5)=0

2. Determine the pivot and free variables. The pivots are in the first and third columns ((x_1) and (x_3). The free variables are (x_2) (x_4) and (x_5)

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

(x_1)=−(x_2)−2*(x_4)+3*(x_5)

(x_3)=5*(x_4)−4*(x_5)

4. Write the general solution vector x by substituting the expressions for (x_1) and (x_3)

x=[[−(x_2)−2*(x_4)+3*(x_5)],[(x_2)],[5*(x_4)−4*(x_5)],[(x_4)],[(x_5)]]

5. Decompose the vector into a linear combination of basis vectors by separating the terms for each free variable:

x=(x_2)*[[−1],[1],[0],[0],[0]]+(x_4)*[[−2],[0],[5],[1],[0]]+(x_5)*[[3],[0],[−4],[0],[1]]

Final Answer

Kernel(A)=span*{[[−1],[1],[0],[0],[0]],[[−2],[0],[5],[1],[0]],[[3],[0],[−4],[0],[1]]}

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