Now that we've established the concepts of vector spaces, span, and linear independence, let's look at two fundamental subspaces intrinsically linked to any given matrix $A$ . These subspaces, the Column Space and the Null Space, provide deep insights into the behavior of the linear transformation defined by $A$ and the structure of the data it might represent. Understanding them is significant for analyzing linear models, feature relationships, and the solvability of linear systems.

The Column Space (Range)

Consider an $m \times n$ matrix $A$ . Its columns, $a_1, a_2, ..., a_n$ , are vectors in $R^m$ .

The Column Space of $A$ , denoted as $Col(A)$ , is the set of all possible linear combinations of the columns of $A$ . In other words, it's the span of the columns: $Col(A) = \text{span}\{a_1, a_2, ..., a_n\}$ Since the span of any set of vectors forms a subspace, $Col(A)$ is a subspace of $R^m$ .

What does the column space represent? Recall that the matrix-vector product $Ax$ can be written as a linear combination of the columns of $A$ with weights given by the entries of $x$ : $Ax = x_1 a_1 + x_2 a_2 + ... + x_n a_n$ This means that the column space $Col(A)$ is precisely the set of all possible outputs $b$ that can be achieved by the transformation $x \mapsto Ax$ . If you think of $A$ as defining a linear transformation $T: R^n \to R^m$ where $T(x) = Ax$ , then $Col(A)$ is the range of this transformation. It's the subspace in the output space $R^m$ that the transformation can actually reach.

A system of linear equations $Ax = b$ is consistent (i.e., has at least one solution) if and only if the vector $b$ is in the column space of $A$ . If $b$ lies outside $Col(A)$ , then it's impossible to find weights $x_1, ..., x_n$ such that $x_1 a_1 + ... + x_n a_n = b$ .

Finding a Basis for the Column Space A basis for the column space consists of a linearly independent set of columns that still span the entire column space. How do we find such a set? The pivot columns of the original matrix $A$ form a basis for $Col(A)$ . To identify them, you typically row reduce $A$ to its echelon form. The columns in the original matrix $A$ that correspond to the pivot positions in the echelon form constitute a basis for $Col(A)$ .

The dimension of the column space, $dim(Col(A))$ , is the number of vectors in its basis, which is equal to the number of pivot columns in $A$ . This dimension has a special name: the rank of the matrix $A$ . $\text{rank}(A) = dim(Col(A))$ The rank tells us the number of independent directions or dimensions spanned by the columns of $A$ . In a machine learning context, if the columns represent features or transformed features, the rank indicates the effective dimensionality of that feature space. A rank less than the number of columns implies some linear dependence or redundancy among the features represented by the columns.

The Null Space (Kernel)

The Null Space of an $m \times n$ matrix $A$ , denoted as $Nul(A)$ , is the set of all vectors $x$ in $R^n$ that are mapped to the zero vector in $R^m$ by the transformation $x \mapsto Ax$ . Formally: $Nul(A) = \{x \in R^n \mid Ax = 0\}$ The null space consists of all solutions to the homogeneous system of linear equations $Ax = 0$ .

Is $Nul(A)$ a subspace? Yes, it's a subspace of the input space $R^n$ . We can verify this:

The zero vector $0 \in R^n$ is always in $Nul(A)$ because $A0 = 0$ .
If $u$ and $v$ are in $Nul(A)$ , then $Au = 0$ and $Av = 0$ . Then $A(u+v) = Au + Av = 0 + 0 = 0$ , so $u+v$ is in $Nul(A)$ .
If $u$ is in $Nul(A)$ and $c$ is a scalar, then $Au = 0$ . Then $A(cu) = c(Au) = c(0) = 0$ , so $cu$ is in $Nul(A)$ .

Geometrically, if $A$ represents a transformation $T: R^n \to R^m$ , the null space $Nul(A)$ is the set of all input vectors that get "squashed" or mapped onto the origin in the output space. This is also sometimes called the kernel of the transformation $T$ .

Finding a Basis for the Null Space To find a basis for the null space, you need to find the general solution to the homogeneous equation $Ax=0$ . This typically involves:

Row reducing the augmented matrix $[A | 0]$ to reduced row echelon form (RREF).
Expressing the basic variables (corresponding to pivot columns) in terms of the free variables (corresponding to non-pivot columns).
Writing the general solution vector $x$ as a linear combination of vectors, where the weights are the free variables. The vectors in this linear combination form a basis for $Nul(A)$ .

The dimension of the null space, $dim(Nul(A))$ , is the number of vectors in its basis, which is equal to the number of free variables in the system $Ax=0$ . This dimension is often called the nullity of the matrix $A$ . $\text{nullity}(A) = dim(Nul(A))$

If the nullity is greater than zero, it means there are non-zero input vectors that map to the zero vector. In terms of the system $Ax=b$ , if a solution $x_p$ exists (a particular solution), then any other solution can be written as $x = x_p + x_h$ , where $x_h$ is any vector from the null space ( $Ax_h = 0$ ). A non-trivial null space implies that solutions to $Ax=b$ , if they exist, are not unique.

The Rank-Nullity Theorem

There's a fundamental relationship connecting the dimension of the column space (rank) and the dimension of the null space (nullity) for any $m \times n$ matrix $A$ . It's known as the Rank-Nullity Theorem: $\text{rank}(A) + \text{nullity}(A) = n$ or $dim(Col(A)) + dim(Nul(A)) = \text{Number of columns of } A$

This theorem provides a powerful link between the input space ( $R^n$ ) and the output space ( $R^m$ ) of the transformation $T(x)=Ax$ . It essentially states that the number of dimensions preserved by the transformation (the rank, dimension of the range) plus the number of dimensions collapsed to zero (the nullity, dimension of the kernel) must equal the total dimension of the input domain ( $n$ ).

Consider what this implies:

If a matrix has a large null space (high nullity), many input dimensions are lost in the transformation, resulting in a lower-dimensional output space (low rank).
If a matrix has a trivial null space (nullity = 0, meaning only $x=0$ solves $Ax=0$ ), then no dimensions are lost, and the rank will be equal to the number of columns ( $n$ ). This happens if and only if the columns of $A$ are linearly independent.

Relevance in Data Analysis and Machine Learning

Why are these subspaces important in practice?

Column Space and Rank: The column space represents the space of all possible outcomes or predictions achievable by a linear model whose coefficients are determined by solving a system involving $A$ . The rank of a feature matrix tells you the effective number of independent features. If the rank is less than the number of columns, it signals multicollinearity or redundant features, which can cause instability in some models (like linear regression). Dimensionality reduction techniques often aim to find a lower-rank approximation that captures most of the structure in $Col(A)$ .
Null Space: The null space identifies combinations of inputs (e.g., feature values) that have no effect on the output of the linear transformation $Ax$ . If $Nul(A)$ contains non-zero vectors, it means different input combinations can lead to the same output. In solving systems like $Ax=b$ , a non-trivial null space indicates infinitely many solutions if one exists. This relates to concepts like parameter identifiability in models. For instance, if the nullity of a design matrix in regression is greater than zero, the model coefficients are not uniquely determined.

By analyzing the column space, null space, and rank of matrices derived from datasets, we gain a deeper understanding of feature relationships, model capabilities, and potential issues like redundancy or non-uniqueness of solutions. These concepts form the bedrock for more advanced techniques explored later, including matrix decompositions like SVD.