Question

Let \(A=\) cr where \(\mathbf{c} \neq \mathbf{0}\) is a column in \(\mathbb{R}^{m}\) and \(\mathbf{r} \neq \mathbf{0}\) is a row in \(\mathbb{R}^{n}\) a. Show that \(\operatorname{col} A=\operatorname{span}\\{\mathbf{c}\\}\) and \(\operatorname{row} A=\operatorname{span}\\{\mathbf{r}\\}\) b. Find \(\operatorname{dim}(\) null \(A)\). c. Show that null \(A=\) null \(\mathbf{r}\).

Short answer

a. \(\operatorname{col} A=\operatorname{span}\{\mathbf{c}\}, \operatorname{row} A=\operatorname{span}\{\mathbf{r}\}.\) b. \(\operatorname{dim}( ext{null } A)=n-1\). c. Null \(A=\) null \(\mathbf{r}\).

Step-by-step solution

Analyze the structure of matrix A

Since \(A\) is formed by the outer product of \(\mathbf{c}\) and \(\mathbf{r}\), it results in a matrix \(A = \mathbf{c} \mathbf{r}\) where \( \mathbf{c} \in \mathbb{R}^m\) and \( \mathbf{r} \in \mathbb{R}^n\). This means \(A\) is an \(m \times n\) matrix where every column is a scaled version of \(\mathbf{c}\) and every row is a scaled version of \(\mathbf{r}\).

Show column space of A

Each column of \(A\) is a scalar multiple of \(\mathbf{c}\), because the outer product repeats the column vector \(\mathbf{c}\) for every entry in \(\mathbf{r}\). Thus, the column space of \(A\), \(\operatorname{col} A\), is the span of \(\{\mathbf{c}\}\).

Show row space of A

Each row of \(A\) is a scalar multiple of \(\mathbf{r}\), because the outer product repeats the row vector \(\mathbf{r}\). Thus, the row space of \(A\), \(\operatorname{row} A\), is the span of \(\{\mathbf{r}\}\).

Determine dimension of null space of A

\(A\) is an \(m \times n\) matrix with rank 1 (since both the column and row spaces are spanned by a single vector). Using the rank-nullity theorem, \[ \operatorname{dim}( ext{null } A) = n - \operatorname{rank}(A) = n - 1. \]

Show null space of A equals null space of r

By definition of the null space, a vector \(\mathbf{x} \in \text{null } A\) means \(A\mathbf{x} = \mathbf{0}\), i.e., \(\mathbf{c} \mathbf{r} \mathbf{x} = \mathbf{0}\). Since \(\mathbf{c} eq \mathbf{0}\), this simplifies to \(\mathbf{r} \mathbf{x} = 0\). Hence, null space of \(A\) is equivalent to null space of \(\mathbf{r}\).

Key concepts

Column Space

The column space of a matrix is a fundamental concept in matrix theory. Essentially, it consists of all possible linear combinations of the matrix's columns. In our example, if matrix \( A \) is constructed as the outer product of vectors \( \mathbf{c} \) and \( \mathbf{r} \), each column is simply a scalar multiple of \( \mathbf{c} \). This scenario arises because the outer product repeats \( \mathbf{c} \) across every entry of \( \mathbf{r} \). Consequently, the column space of matrix \( A \), denoted as \( \operatorname{col} A \), is spanned by the single vector \( \mathbf{c} \). Here are some key points regarding the column space:

• The column space of matrix \( A \) is the set of all vectors that can be expressed as linear combinations of the columns of \( A \).
• If you're given a matrix \( A = \mathbf{c} \mathbf{r} \), \( \operatorname{col} A \) is simply \( \operatorname{span}\{\mathbf{c}\} \).
• In practical terms, knowing the column space can help us understand the solutions to the equation \( A\mathbf{x} = \mathbf{b} \).

Understanding the column space allows for greater insight into the dimensions and rank of the matrix, both crucial in applications such as determining linear dependence of vectors, solving systems of equations, and more.

Row Space

The row space of a matrix, much like the column space, is a key aspect of linear algebra. In simple terms, it involves all possible linear combinations of the rows of the matrix. For our given matrix \( A \), which is the outer product of \( \mathbf{c} \) and \( \mathbf{r} \), each row is a scalar multiple of \( \mathbf{r} \). This happens because when forming the matrix \( A \) by the outer product, \( \mathbf{r} \) is repeated across all rows influnecing every column. Therefore, the row space of \( A \), denoted as \( \operatorname{row} A \), is spanned by the vector \( \mathbf{r} \). Here are some practical insights about the row space:

• The row space of a matrix is essentially the span of its row vectors.
• For matrix \( A = \mathbf{c} \mathbf{r} \), \( \operatorname{row} A \) is equivalent to \( \operatorname{span}\{\mathbf{r}\} \).
• Understanding the row space helps analyze the nature of solutions to matrix equations and transformation properties.

The row space aids in understanding transitions between different vector spaces and has important implications in determining matrix rank and solving equations involving matrices.

Null Space

In matrix theory, the null space is a concept that carries significant value. It is defined as the set of all vectors \( \mathbf{x} \) that satisfy the equation \( A\mathbf{x} = \mathbf{0} \). In the context of our problem, the matrix \( A \) is formed by the outer product of \( \mathbf{c} \) and \( \mathbf{r} \), and the null space is closely related to \( \mathbf{r} \). The logic is as follows: any vector \( \mathbf{x} \) in the null space of \( A \) must fulfill \( \mathbf{c} \mathbf{r} \mathbf{x} = \mathbf{0} \). Given that \( \mathbf{c} eq \mathbf{0} \), it simplifies to \( \mathbf{r} \mathbf{x} = 0 \). Thus, the null space of matrix \( A \) coincides with the null space of \( \mathbf{r} \). Some important characteristics of the null space include:

• The dimension of the null space provides insights into the solutions of the homogeneous equation \( A\mathbf{x} = \mathbf{0} \).
• For matrix \( A = \mathbf{c} \mathbf{r} \), since the rank of the matrix is 1, the dimension of the null space is \( n - 1 \) where \( n \) is the number of columns in \( A \).
• The null space of \( A \) being equivalent to the null space of \( \mathbf{r} \) means solving \( A\mathbf{x} = 0 \) is identical to solving \( \mathbf{r}\mathbf{x} = 0 \).

The null space is essential for understanding the solution space of linear equations and provides information about the degrees of freedom in a system represented by a matrix.