Problem 4 - Differential Equations and Linear Algebra

Question

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{r} -3 \\ 1 \\ 7 \end{array}\right]$$

Step-by-Step Solution

Answer

The row space of matrix \(A\) is a subspace of \(\mathbb{R}^n\) where \(n=1\), and its basis is \(\{(-3)\}\). The column space of matrix \(A\) is a subspace of \(\mathbb{R}^m\) where \(m=3\), and its basis is \(\left\{ \begin{pmatrix} -3 \\ 1 \\ 7 \end{pmatrix}\right\}\).

Step 1: (a) Determine the value of n and the basis for the row space of A

Since \(A\) is a \(3\times 1\) matrix (a single column with three elements), there is only one row in the matrix. Therefore, the row space of \(A\) will be a subspace of \(\mathbb{R}^n\) where \(n=1\). The basis for the row space of \(A\) is the single row in the matrix, so the basis is: \( \text{Basis of row space}(A) = \{(-3)\} \)

Step 2: (b) Determine the value of m and the basis for the column space of A

Since \(A\) is a \(3\times 1\) matrix, it has 3 rows and 1 column. The column space of \(A\) is a subspace of \(\mathbb{R}^m\) where \(m\) is the number of rows in the matrix. In this case, \(m=3\). The basis for the column space of \(A\) is formed by the single column in the matrix, so the basis is: \( \text{Basis of col space}(A) = \left\{ \begin{pmatrix} -3 \\ 1 \\ 7 \end{pmatrix}\right\} \)

Key Concepts

Row Space

The row space of a matrix is formed by the linear combinations of its rows. Consider a matrix \( A \) with dimensions \( m \times n \). This means \( A \) has \( m \) rows and \( n \) columns. The row space is a subspace of \( \mathbb{R}^n \) and consists of all possible sums that can be formed using the rows of the matrix as vectors.

To determine the row space, particularly its dimension and a basis, we typically examine the row vectors themselves. In the current problem, matrix \( A \) is a \( 3 \times 1 \) matrix, which means it has only one row vector. In this case, that's why the row space is simply spanned by this single row vector, making it a subspace of \( \mathbb{R}^1 \). The single row of matrix \( A \), \( \{-3\} \), is itself a basis for the row space.

• Row space consists of all linear combinations of the row vectors.
• The dimension of the row space depends on the number of linearly independent rows.
• For a \( 3 \times 1 \) matrix, the row space is trivial and is spanned by its single row.

Column Space

When discussing the column space, we are looking at a different perspective of a matrix. The column space is the span of the matrix's column vectors. For a matrix \( A \), with dimensions \( m \times n \), the column space is considered a subspace of \( \mathbb{R}^m \).

In our particular case, matrix \( A \) is again a \( 3 \times 1 \) matrix. It comprises one column, so the column space is spanned by the single column vector. This means the column space is a subspace of \( \mathbb{R}^3 \), given that the matrix has three rows. The basis of the column space is simply \( \begin{pmatrix} -3 \ 1 \ 7 \end{pmatrix} \), the vector itself.

• Column space consists of all linear combinations of the columns.
• The dimension of the column space indicates the rank of the matrix.
• For a \( 3 \times 1 \) matrix, its single column neatly represents the entire column space.

Matrix Subspaces

Matrix subspaces are crucial elements in linear algebra, representing various spans formed by the matrix's rows or columns. A subspace, by definition, is a portion of a vector space that is closed under vector addition and scalar multiplication.

There are two primary subspaces we often consider: the row space and the column space. These subspaces form the row and column vectors, respectively. Given our matrix \( A \), we see that:

• The row space of \( A \) is a subspace of \( \mathbb{R}^n \), where \( n \) is the number of columns.
• The column space of \( A \) is a subspace of \( \mathbb{R}^m \), where \( m \) is the number of rows.
• Each of these spaces has a basis, which is a minimal set of vectors needed to span the subspace.
• Knowing these subspaces aids in solving systems of equations and understanding the matrix's properties.

Basis

A basis in linear algebra is a set of vectors that are linearly independent and span a vector space or subspace. This concept helps identify the dimension of that space and provides a framework to express any vector within that space as a combination of the basis vectors.

The basis of a subspace is crucial because it represents the 'building blocks' of the space. For example, given our matrix \( A \):

• The row space's basis is the single row of the matrix, as it's the only vector needed to describe all possible row combinations.
• The column space is similarly defined by its columns; with only one column, the entire space is represented by this single column vector.
• A determinant feature of a basis is its requirement of linear independence, ensuring no vector in the set can be expressed as a combination of the others.
• Understanding the basis is essential for computations and transformations in linear algebra.