Column Space of a Matrix

Linear Algebra is a broad topic of mathematics with applications in various real-world situations, particularly in machine learning. Matrices and vectors are the fundamental building blocks of linear algebra, and they are used in a variety of procedures and tools. The column space of a matrix will be discussed in this article. We will also go over several necessary terminologies for comprehending the matrix’s column space.

What is the Span of a Vector?

Span simply means that given a set of vectors, if any linear combination is applied to that set of vectors and it remains within that vector space, it spans that vector space. This means that if you multiply any scalar by a specific vector, it will remain within that dimension, whether you are working with the first, second, third, or nth dimension. It is said that it “spans” everywhere within that dimension. When you multiply a set of vectors by a scalar, it simply indicates that the set of vectors you are working with can cover (or be placed anywhere inside) the full dimension (or vector space) you are working with.

What is Linear Combination?

Suppose you have a set of mathematical objects {x₁….x_n} that support scalar multiplication and addition (e.g., members of a ring or a vector space), then y = a₁x₁+a₂x₂+… a_nx_n (where ai are some scalars values). The most popular illustration is to utilize 3D vectors in Euclidean space. A vector that resides in the same plane through the origin as the original two vectors put at the origin is a linear combination of any two such vectors.

What are Row and Column Spaces?

Assume A is an mxn matrix over the field F. Then there are n-component vectors in the rows, and there are m of them. Similarly, each m-component vector is represented by n columns. The subspace of Fⁿ formed by the row vectors is A’s row-space, and its elements are linear combinations of the row vectors. This space has dimension, and the columns compel such relationships between the rows and vice versa. Similarly, the matrix’s column-space is the subspace of F^m formed by the matrix’s column vectors. Although this space is distinct from row space in general, it has the same dimensions as row space since any linear relationship between the columns also imposes such relations among the rows and vice versa.

Diving more into the Column Space

Span is the more fundamental concept. Simply put, the span of the columns of a given vector is what we call the column space. You can take all possible linear combinations of vectors if you have a collection of them. The resulting vector space is known as the span of the original collection. The column space is a collection of a set of all possible linear combinations of the matrix’s column vectors. In other words, if a vector b in R^m can be expressed as a linear combination of A’s columns, it is in A’s column space. That is, b ∈ CS(A) precisely when there exist scalars x₁, x₂, …, x_n such that

As the product of A with a column vector, any linear combination of the column vectors of a matrix A can be written:

Therefore, the column space of matrix A consists of all possible products A*x, for x ∈ Cⁿ. The above result is also the image of the corresponding matrix transformation.

We usually denote the row and column spaces of the matrix (let us say A) by C(AT) and C(A), respectively.

Conclusion

This article covered various topics relating to the matrix’s column space. The span of a vector is the space that stays unchanged after a linear combination is applied to the collection of vectors. After multiplying a set of vectors and scalars, the summation is called a linear combination. The collection of all conceivable linear combinations of a matrix’s column vectors is the matrix’s column space.