Determine the dimension of the null space of A = [[1,2,3],[4,5,6],[7,8,9]].

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Determine the dimension of the null space of A = [[1,2,3],[4,5,6],[7,8,9]].

The dimension of the null space is the number of columns minus the rank of the matrix (nullity = n − rank). Here n = 3, so we need the rank. The third column is a combination of the first two: [3, 6, 9] = 2[2, 5, 8] − [1, 4, 7], so the columns are not all independent and the rank is at most 2. The first two columns are not multiples of each other, so they are independent, giving rank = 2. Therefore the nullity is 3 − 2 = 1. A basis for the null space is given by the vector (1, −2, 1), since A(1, −2, 1)ᵀ = 0, and all solutions are multiples of this vector.

The dimension of the null space is the number of columns minus the rank of the matrix (nullity = n − rank). Here n = 3, so we need the rank.

The third column is a combination of the first two: [3, 6, 9] = 2[2, 5, 8] − [1, 4, 7], so the columns are not all independent and the rank is at most 2. The first two columns are not multiples of each other, so they are independent, giving rank = 2.

Therefore the nullity is 3 − 2 = 1. A basis for the null space is given by the vector (1, −2, 1), since A(1, −2, 1)ᵀ = 0, and all solutions are multiples of this vector.

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