Final answer:
The null space of a matrix consists of all the vectors in the vector space that when multiplied by the matrix result in the zero vector. To find a non-zero vector in the null space of a matrix, you can find the reduced row-echelon form (RREF) of the matrix and solve for the leading variables using the free variables.
Step-by-step explanation:
The null space of a matrix consists of all the vectors in the vector space that when multiplied by the matrix result in the zero vector. In other words, it represents the solutions to the homogeneous equation Ax = 0, where A is the matrix and x is a vector.
To find a non-zero vector in the null space of a matrix, you can first find the reduced row-echelon form (RREF) of the matrix. The pivot columns in the RREF will correspond to the leading variables, while the non-pivot columns will correspond to the free variables.
For each free variable, assign a value to it (except for 0) and solve for the corresponding leading variables using the equations in the RREF. The resulting solutions will give you the non-zero vectors in the null space of the matrix.