# Chapter 26. Null Space # Chapter 26. Null Space The null space of a matrix is the set of all vectors that the matrix sends to zero. If \(A\) is an \(m \times n\) matrix, then its null space is the solution set of the homogeneous system $$ Ax=0. $$ Equivalently, the null space is the kernel of the linear transformation \(x \mapsto Ax\). It is always a subspace of the domain \(F^n\). ## 26.1 Definition Let \(A\) be an \(m \times n\) matrix over a field \(F\). The null space of \(A\), denoted $$ \operatorname{Null}(A), $$ is defined by $$ \operatorname{Null}(A) = \{x \in F^n : Ax=0\}. $$ The vectors in the null space are the solutions of a homogeneous linear system. Since \(A\) has \(n\) columns, the input vector \(x\) has \(n\) components. Therefore $$ \operatorname{Null}(A) \subseteq F^n. $$ The null space lives in the domain of the matrix transformation, not in the codomain. ## 26.2 Homogeneous Systems A system is homogeneous when its right-hand side is zero: $$ Ax=0. $$ Such a system is always consistent because $$ x=0 $$ is always a solution. This solution is called the trivial solution. The main question is whether there are nontrivial solutions: $$ x \neq 0. $$ Nontrivial solutions exist exactly when the columns of \(A\) are linearly dependent. ## 26.3 Null Space as a Subspace The null space is a subspace of \(F^n\). First, it contains the zero vector because $$ A0=0. $$ Next, let $$ u,v \in \operatorname{Null}(A). $$ Then $$ Au=0 $$ and $$ Av=0. $$ Therefore $$ A(u+v)=Au+Av=0+0=0. $$ So $$ u+v \in \operatorname{Null}(A). $$ For a scalar \(c\), $$ A(cu)=cAu=c0=0. $$ So $$ cu \in \operatorname{Null}(A). $$ Thus \(\operatorname{Null}(A)\) is closed under addition and scalar multiplication. ## 26.4 Geometric Meaning The null space contains all directions that are collapsed to zero by the transformation $$ x \mapsto Ax. $$ If $$ x \in \operatorname{Null}(A), $$ then $$ Ax=0. $$ Thus \(x\) is invisible to the transformation: adding \(x\) to an input does not change the output. Indeed, if \(Ax=0\), then for any vector \(v\), $$ A(v+x)=Av+Ax=Av. $$ So null-space vectors are directions of ambiguity. ## 26.5 Example in \(\mathbb{R}^2\) Let $$ A= \begin{bmatrix} 1 & 2 \end{bmatrix}. $$ Then $$ Ax=0 $$ means $$ \begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} =0. $$ So $$ x+2y=0. $$ Solving for \(x\), $$ x=-2y. $$ Let $$ y=t. $$ Then $$ x=-2t. $$ Thus $$ \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} -2t\\ t \end{bmatrix} = t \begin{bmatrix} -2\\ 1 \end{bmatrix}. $$ Therefore $$ \operatorname{Null}(A) = \operatorname{span} \left( \begin{bmatrix} -2\\ 1 \end{bmatrix} \right). $$ This null space is a line through the origin in \(\mathbb{R}^2\). ## 26.6 Example in \(\mathbb{R}^3\) Let $$ A= \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}. $$ The equation $$ Ax=0 $$ becomes $$ x_1+x_2+x_3=0. $$ Solve for \(x_1\): $$ x_1=-x_2-x_3. $$ Let $$ x_2=s, \qquad x_3=t. $$ Then $$ x= \begin{bmatrix} -s-t\\ s\\ t \end{bmatrix} = s \begin{bmatrix} -1\\ 1\\ 0 \end{bmatrix} + t \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix}. $$ Therefore $$ \operatorname{Null}(A) = \operatorname{span} \left( \begin{bmatrix} -1\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix} \right). $$ The null space is a plane through the origin in \(\mathbb{R}^3\). ## 26.7 Finding the Null Space To find \(\operatorname{Null}(A)\), solve $$ Ax=0. $$ The standard procedure is: | Step | Operation | |---|---| | 1 | Row reduce \(A\) to echelon or reduced echelon form | | 2 | Identify pivot variables and free variables | | 3 | Express pivot variables in terms of free variables | | 4 | Write the solution in parametric vector form | | 5 | Extract a basis from the parameter vectors | The parameter vectors form a basis for the null space. ## 26.8 Row Reduction Example Let $$ A= \begin{bmatrix} 1 & 2 & 1 & 0\\ 2 & 4 & 3 & 1\\ 1 & 2 & 2 & 1 \end{bmatrix}. $$ Solve $$ Ax=0. $$ Row reduce: $$ \begin{bmatrix} 1 & 2 & 1 & 0\\ 2 & 4 & 3 & 1\\ 1 & 2 & 2 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 \end{bmatrix}. $$ The equations are $$ x_1+2x_2+x_3=0, $$ $$ x_3+x_4=0. $$ The pivot variables are \(x_1\) and \(x_3\). The free variables are \(x_2\) and \(x_4\). Let $$ x_2=s, \qquad x_4=t. $$ Then $$ x_3=-t. $$ From the first equation, $$ x_1+2s-t=0, $$ so $$ x_1=-2s+t. $$ Thus $$ x= \begin{bmatrix} -2s+t\\ s\\ -t\\ t \end{bmatrix} = s \begin{bmatrix} -2\\ 1\\ 0\\ 0 \end{bmatrix} + t \begin{bmatrix} 1\\ 0\\ -1\\ 1 \end{bmatrix}. $$ Therefore a basis for the null space is $$ \left( \begin{bmatrix} -2\\ 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 1\\ 0\\ -1\\ 1 \end{bmatrix} \right). $$ So $$ \dim \operatorname{Null}(A)=2. $$ ## 26.9 Nullity The nullity of a matrix is the dimension of its null space: $$ \operatorname{nullity}(A) = \dim \operatorname{Null}(A). $$ In the previous example, $$ \operatorname{nullity}(A)=2. $$ Nullity counts the number of independent directions that \(A\) sends to zero. It also equals the number of free variables in the homogeneous system \(Ax=0\). ## 26.10 Rank-Nullity If \(A\) is an \(m \times n\) matrix, then $$ \operatorname{rank}(A)+\operatorname{nullity}(A)=n. $$ The number \(n\) is the dimension of the domain \(F^n\). Rank counts the number of pivot variables. Nullity counts the number of free variables. Since each column corresponds to one variable, $$ \text{pivot variables}+\text{free variables}=n. $$ This theorem is one of the central counting principles of linear algebra. ## 26.11 Null Space and Linear Independence The columns of \(A\) are linearly independent exactly when $$ \operatorname{Null}(A)=\{0\}. $$ To see this, write $$ A= \begin{bmatrix} |&|&&|\\ a_1&a_2&\cdots&a_n\\ |&|&&| \end{bmatrix}. $$ Then $$ Ax=x_1a_1+x_2a_2+\cdots+x_na_n. $$ The equation $$ Ax=0 $$ is exactly the equation $$ x_1a_1+x_2a_2+\cdots+x_na_n=0. $$ Thus nonzero solutions correspond to nontrivial linear relations among the columns. Therefore: | Null space | Columns of \(A\) | |---|---| | \(\operatorname{Null}(A)=\{0\}\) | Linearly independent | | \(\operatorname{Null}(A)\neq\{0\}\) | Linearly dependent | ## 26.12 Null Space and Injectivity The linear map $$ T(x)=Ax $$ is injective exactly when $$ \operatorname{Null}(A)=\{0\}. $$ If \(T\) is injective, then only \(0\) can map to \(0\). Conversely, suppose $$ \operatorname{Null}(A)=\{0\}. $$ If $$ Au=Av, $$ then $$ A(u-v)=0. $$ Thus $$ u-v \in \operatorname{Null}(A). $$ So $$ u-v=0, $$ and hence $$ u=v. $$ Therefore \(T\) is injective. The null space measures the failure of injectivity. ## 26.13 Null Space and Information Loss A matrix transformation may lose information. If $$ Ax=0 $$ for some nonzero \(x\), then the transformation cannot distinguish between \(v\) and \(v+x\), because $$ A(v+x)=Av. $$ The null space contains exactly the directions along which information is lost. If the null space is large, many different inputs have the same output. If the null space is trivial, different inputs have different outputs. ## 26.14 Null Space and Solution Sets Consider a nonhomogeneous system $$ Ax=b. $$ If \(x_p\) is one particular solution, then every solution has the form $$ x=x_p+z, $$ where $$ z\in \operatorname{Null}(A). $$ Indeed, if $$ Ax_p=b $$ and $$ Az=0, $$ then $$ A(x_p+z)=Ax_p+Az=b. $$ Conversely, if \(x\) is any solution, then $$ A(x-x_p)=Ax-Ax_p=b-b=0. $$ Thus $$ x-x_p\in \operatorname{Null}(A). $$ Therefore the complete solution set is $$ x_p+\operatorname{Null}(A). $$ The null space gives the homogeneous part of every linear system. ## 26.15 Affine Solution Spaces When \(b\neq 0\), the solution set of $$ Ax=b $$ is generally not a subspace. It may fail to contain the zero vector. Instead, if it is nonempty, it is a translate of the null space: $$ x_p+\operatorname{Null}(A). $$ Such a set is called an affine subspace. For example, a line not passing through the origin is not a vector subspace, but it can be a translate of a one-dimensional null space. Thus homogeneous systems give subspaces. Nonhomogeneous systems give affine spaces when they are consistent. ## 26.16 Null Space of Special Matrices Some matrices have immediate null spaces. | Matrix | Null space | |---|---| | Zero matrix \(0_{m\times n}\) | All of \(F^n\) | | Identity matrix \(I_n\) | \(\{0\}\) | | Invertible square matrix \(A\) | \(\{0\}\) | | Projection matrix | Directions projected away | | Matrix with repeated columns | Contains nonzero relations among those columns | For the zero matrix, $$ 0x=0 $$ for every \(x\), so every vector lies in the null space. For the identity matrix, $$ I x=x, $$ so $$ Ix=0 $$ only when $$ x=0. $$ ## 26.17 Left Null Space For an \(m\times n\) matrix \(A\), the left null space is $$ \operatorname{Null}(A^T). $$ Since \(A^T\) is an \(n\times m\) matrix, $$ \operatorname{Null}(A^T)\subseteq F^m. $$ It consists of all vectors \(y\in F^m\) such that $$ A^Ty=0. $$ Equivalently, $$ y^TA=0. $$ The left null space contains vectors orthogonal to every column of \(A\). Later, this will be expressed as $$ \operatorname{Null}(A^T)=\operatorname{Col}(A)^\perp $$ when an inner product is available. ## 26.18 Null Space and Orthogonality For real matrices with the standard dot product, $$ \operatorname{Null}(A) $$ is orthogonal to the row space of \(A\). Indeed, \(Ax=0\) means each row of \(A\) has dot product zero with \(x\). If the rows are $$ r_1,\ldots,r_m, $$ then $$ r_i\cdot x=0 $$ for every \(i\). Since \(x\) is orthogonal to every row, it is orthogonal to every linear combination of the rows. Thus $$ \operatorname{Null}(A)=\operatorname{Row}(A)^\perp. $$ This relationship belongs to the four fundamental subspaces. ## 26.19 Common Mistakes A common mistake is to place the null space in the wrong ambient space. If \(A\) is \(m\times n\), then $$ Ax $$ is defined for $$ x\in F^n. $$ Therefore $$ \operatorname{Null}(A)\subseteq F^n. $$ The column space, by contrast, lies in \(F^m\). Another common mistake is to confuse the null space with the set of zero columns. The null space is a set of input vectors \(x\), not a set of columns of \(A\). A third mistake is to forget that the null space is always a subspace, while the solution set of \(Ax=b\) with \(b\neq0\) is generally only an affine translate. ## 26.20 Summary The null space of a matrix is the set of all input vectors sent to zero. It is the solution space of the homogeneous system \(Ax=0\), and it is a subspace of the domain. The key ideas are: | Concept | Meaning | |---|---| | Null space | \(\{x:Ax=0\}\) | | Kernel | Another name for null space of a linear map | | Trivial solution | \(x=0\) | | Nontrivial solution | A nonzero vector in the null space | | Nullity | Dimension of the null space | | Rank-nullity | \(\operatorname{rank}(A)+\operatorname{nullity}(A)=n\) | | Injectivity test | \(A\) is injective exactly when \(\operatorname{Null}(A)=\{0\}\) | | General solution | \(x_p+\operatorname{Null}(A)\) | | Orthogonal relation | \(\operatorname{Null}(A)=\operatorname{Row}(A)^\perp\) | The null space records the directions lost by a linear transformation. It explains free variables, nonunique solutions, column dependence, and the geometry of homogeneous systems.