Question

Assume that \(\mathbf{X}\) is an \(n \times p\) matrix. Then the kernel of \(\mathbf{X}\) is defined to be the space \(\operatorname{ker}(\mathbf{X})=\\{\mathbf{b}: \mathbf{X} \mathbf{b}=\mathbf{0}\\}\). (a) Show that \(\operatorname{ker}(\mathbf{X})\) is a subspace of \(R^{p}\). (b) The dimension of \(\operatorname{ker}(\mathbf{X})\) is called the nullity of \(\mathbf{X}\) and is denoted by \(\nu(\mathbf{X})\). Let \(\rho(\mathbf{X})\) denote the rank of \(\mathbf{X}\). A fundamental theorem of linear algebra says that \(\rho(\mathbf{X})+\nu(\mathbf{X})=p .\) Use this to show that if \(\mathbf{X}\) has full column rank, then \(\operatorname{ker}(\mathbf{X})=\\{\mathbf{0}\\}\)

Short answer

In conclusion: (a) The kernel of a matrix \(\mathbf{X}\), \(\operatorname{ker}(\mathbf{X})\), is a subspace of \(R^p\) as the zero vector is in the kernel, the kernel is closed under addition and scalar multiplication. (b) If a matrix \(\mathbf{X}\) has full column rank, then the Rank-Nullity theorem dictates that there are no non-zero vectors in its kernel, hence the kernel only contains the zero vector.

Step-by-step solution

Proving the kernel is a subspace

To show the kernel is a subspace of \(R^p\), one needs to check for three properties (1) The zero vector is in the kernel. (2) The kernel is closed under addition. (3) The kernel is closed under scalar multiplication. (1) Since \(\mathbf{X} \times \mathbf{0} = \mathbf{0}\), the zero vector is in the kernel. (2) For any two vectors \(\mathbf{b_1}, \mathbf{b_2} \in \operatorname{ker}(\mathbf{X})\), their sum \(\mathbf{b_1}+\mathbf{b_2}\) also belongs to the kernel because \(\mathbf{X}(\mathbf{b_1} + \mathbf{b_2}) = \mathbf{X}\mathbf{b_1} + \mathbf{X}\mathbf{b_2} = \mathbf{0} + \mathbf{0} = \mathbf{0}\). (3) Similarly, for scalar \(c\) and for all \(\mathbf{b} \in \operatorname{ker}(\mathbf{X})\), \(c\mathbf{b}\) is also in the kernel as \(\mathbf{X}(c\mathbf{b}) = c(\mathbf{X}\mathbf{b}) = c\mathbf{0} = \mathbf{0}\). Thus, the kernel is a subspace of \(R^p\).

Applying the Rank-Nullity theorem

The Rank-Nullity theorem states that the rank of a matrix plus its nullity is equal to the number of columns in the matrix, i.e., \(\rho(\mathbf{X}) + \nu(\mathbf{X}) = p\). If the matrix \(\mathbf{X}\) has full column rank, then \(\rho(\mathbf{X}) = p\), which implies that \(\nu(\mathbf{X}) = 0\). The nullity being zero means there are no non-zero vectors in the kernel of \(\mathbf{X}\), hence the kernel only contains the zero vector.