Forming a Matrix from a List of Vectors

From an Ordered List to a Matrix

A list of vectors can be organized into a rectangular array without changing any of its entries. The key choice is where each vector goes: to form a matrix from a list, place the vectors as its columns, in the same order as the list. This convention keeps the role of each vector visible and gives a reliable way to check both the entries and the dimensions of the matrix.

Throughout this tutorial, vectors are columns in \(\mathbb{R}^m\), and all entries and scalars are real. An ordered list records not only which vectors occur but also their positions. For example, \((\mathbf{v}_1,\mathbf{v}_2)\) and \((\mathbf{v}_2,\mathbf{v}_1)\) have the same members but a different order. When forming a matrix, that order determines the order of its columns.

Two separate counts determine the dimensions. Each vector in \(\mathbb{R}^m\) has \(m\) coordinates, so it contributes \(m\) entries down one column. The list has \(p\) vectors, so the matrix has \(p\) columns. Thus, if the list contains \(p\) vectors from \(\mathbb{R}^m\), the resulting matrix has \(m\) rows and \(p\) columns. The number of coordinates is not the number of vectors, and these counts need not be equal.

Forming the Columns Carefully

The entries are copied, not combined. For instance, the \(-3\) in the top entry of the third column comes from the first coordinate of \(\mathbf{v}_3\); it is not calculated from the other entries. The same applies to negative entries and zeros. A zero entry remains in its coordinate position just like any other entry.

This example makes the two dimension counts especially visible: four vectors produce four columns, while three coordinates per vector produce three rows. The matrix does not need to be square. A square matrix occurs only when the number of vectors equals the number of coordinates of each vector.

Why the Dimensions and Order Are Forced

Once the list is given, there is no freedom to choose the entries or rearrange the columns. The coordinates specify every entry, and the list positions specify which column receives each vector. The following result states precisely what this construction guarantees.

Proof. Write the coordinates of each vector as \(\mathbf{v}_j=(v_{1j},\dots,v_{mj})^{\top}\). Construct the matrix \(\mathbf{A}=[a_{ij}]\) by setting \(a_{ij}=v_{ij}\) for every row index \(i\in\{1,\dots,m\}\) and column index \(j\in\{1,\dots,p\}\). This defines an \(m\times p\) matrix, and its \(j\)th column is \((v_{1j},\dots,v_{mj})^{\top}=\mathbf{v}_j\). Thus such a matrix exists. For uniqueness, suppose \(\mathbf{D}=[d_{ij}]\) is another \(m\times p\) matrix with the same columns. Equality of its \(j\)th column with \(\mathbf{v}_j\) gives \(d_{ij}=v_{ij}\) for every \(i\) and \(j\). Hence \(d_{ij}=a_{ij}\) for every entry, so \(\mathbf{D}=\mathbf{A}\). \(\square\)

The dimensions in the theorem follow from the same indexing: \(i\) ranges over the \(m\) coordinates within a vector, while \(j\) ranges over the \(p\) positions in the list. This accounts for every entry and leaves no additional row or column to add. It also explains why the formation rule does not depend on the particular coordinate values: changing an entry changes the corresponding matrix entry, but it does not change the dimensions.

Common Mistake: Writing the Vectors as Rows

A frequent error is to place the coordinates of each vector across a row. That creates a matrix whose rows, rather than columns, are the given vectors. For example, for \(\mathbf{v}_1=\begin{bmatrix}2\\-1\end{bmatrix}\), \(\mathbf{v}_2=\begin{bmatrix}0\\5\end{bmatrix}\), and \(\mathbf{v}_3=\begin{bmatrix}-3\\4\end{bmatrix}\), the following is a wrong formation if the instruction is to place the vectors as columns:

It has three rows and two columns, and its rows are \((2,-1)\), \((0,5)\), and \((-3,4)\). Its first column is \(\begin{bmatrix}2\\0\\-3\end{bmatrix}\), which is not \(\mathbf{v}_1\), so it fails the column check. The correction is to place each vector vertically as a column:

This corrected matrix has two rows and three columns, and reading down its columns recovers the original list. A second possible mistake is to put the correct vectors into the wrong order. That also fails the check: the \(j\)th column must match the \(j\)th vector, not merely one of the vectors somewhere in the list.

Why This Organization Is Useful

Placing a list of vectors into one matrix gives a compact way to record several vectors while preserving their identities and order. It also makes dimension checks immediate: a list of \(p\) vectors in \(\mathbb{R}^m\) produces an \(m\times p\) matrix. In later work, matrix columns will be used to express operations involving the whole list at once. For now, the essential skill is simply to build the matrix accurately and verify it by recovering the original vectors from its columns.

The construction imposes no condition that the vectors be different, nonzero, or independent. If two vectors in the list happen to have identical coordinates, their corresponding columns are identical. If a vector is the zero vector, its column consists entirely of zeros. These features do not change the rule or the dimensions; the matrix records the list as given.