Question

Let \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) be a linear operator of rank \(1,\) where \(\mathbb{R}^{n}\) is written as rows. Show that there exist numbers \(a_{1}, a_{2}, \ldots, a_{n}\) and \(b_{1}, b_{2}, \ldots, b_{n}\) such that \(T(X)=X A\) for all rows \(X\) in \(\mathbb{R}^{n},\) where $$ A=\left[\begin{array}{cccc} a_{1} b_{1} & a_{1} b_{2} & \cdots & a_{1} b_{n} \\ a_{2} b_{1} & a_{2} b_{2} & \cdots & a_{2} b_{n} \\ \vdots & \vdots & & \vdots \\ a_{n} b_{1} & a_{n} b_{2} & \cdots & a_{n} b_{n} \end{array}\right] $$ \(\left[\right.\) Hint \(: \operatorname{im} T=\mathbb{R} \mathbf{w}\) for \(\mathbf{w}=\left(b_{1}, \ldots, b_{n}\right)\) in \(\left.\mathbb{R}^{n} .\right]\)

Short answer

Numbers \( a_i \) and \( b_i \) exist such that \( T(X) = XA \).

Step-by-step solution

Understanding the Problem Statement

We are given a linear operator \( T: \mathbb{R}^n \rightarrow \mathbb{R}^n \) of rank 1. Our goal is to demonstrate that there are numbers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \) such that \( T(X) = XA \) for all \( X \) in \( \mathbb{R}^n \), where \[ A = \begin{bmatrix} a_1b_1 & a_1b_2 & \cdots & a_1b_n \ a_2b_1 & a_2b_2 & \cdots & a_2b_n \ \vdots & \vdots & & \vdots \ a_nb_1 & a_nb_2 & \cdots & a_nb_n \end{bmatrix} \]. The hint suggests the image of \( T \) is spanned by a vector \( \mathbf{w} = (b_1, \ldots, b_n) \).

Identify the structure of a rank 1 operator

A rank 1 linear operator means that the image of the operator is one-dimensional. This implies that for any vector \( X \) in \( \mathbb{R}^n \), \( T(X) \) can be expressed as a scalar multiple of a single vector \( \mathbf{w} = (b_1, b_2, \ldots, b_n) \). Thus, \( T(X) = c X\mathbf{w} \) for some scalar \( c \).

Express \( T \) using a matrix \( A \)

Given \( T(X) = X\mathbf{w} \), we want to find a matrix \( A \) such that \( XA = X\mathbf{w} \) for all \( X \). This implies that the columns of \( A \) are multiples of \( \mathbf{w} \). If \( \mathbf{w} = (b_1, b_2, \ldots, b_n) \), a suitable choice is making each column of \( A \) consist of \( (a_1 b_j, a_2 b_j, \ldots, a_n b_j) \) for columns indexed by \( j \).

Construct the matrix \( A \) explicitly

Define the rows of \( A \) as \( (a_i b_1, a_i b_2, \ldots, a_i b_n) \) for each index \( i \), with \( a_i \) chosen for simplicity as the coefficient in the linear combination forming the image under \( T \). Specifically, write \( a_i = \text{projection factor for } X \, \text{in } \, T(X) \).

Verify the expression for \( A \) satisfies the problem's condition

Express \( XA \) using the defined rows of \( A \), this yields each row of the resulting matrix multiplying \( b_1, b_2, \ldots, b_n \) by corresponding constants \( a_1, a_2, \ldots, a_n \). Thus, \( XA = T(X) \) for all vectors \( X \) in \( \mathbb{R}^n \), confirming the expression of \( A \).

Conclusion

Therefore, numbers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \) exist such that \( T(X)=X A \) where \( A \) has the structure described in the problem, for the linear operator of rank 1 with the specified image.

Key concepts

Rank of a Matrix

In linear algebra, the rank of a matrix provides essential information about the matrix's properties and capabilities. Specifically, the rank is the maximum number of linearly independent row vectors in the matrix. Alternatively, it is the same as the number of linearly independent column vectors.

• When we say a matrix has rank 1, it means all its rows or columns can be viewed as scalar multiples of a single vector.
• This reflects that the matrix effectively "compresses" information into a one-dimensional space.

Given the exercise, the linear operator is of rank 1. Therefore, any vector in the image of the transformation can be represented as a scalar multiplication of a single vector. This profound characteristic simplifies many calculations and expressions in linear transformations. It also simplifies the matrix representation of operators in scenarios like those presented in the problem.

Matrix Multiplication

Matrix multiplication is a fundamental operation crucial in expressing linear transformations. It is defined in such a way that the dimensions of the matrices involved match appropriately: if matrix A is of size \(m \times n\), it can only be multiplied by a matrix B of size \(n \times p\).
The multiplication result will be a matrix of size \(m \times p\). For each element in the resulting matrix, you calculate the dot product of corresponding row and column elements. This operation is neatly exemplified in the exercise.

• When transforming a vector via a matrix \(A\), the matrix multiplication \(T(X) = XA\) shows how each component of \(X\) interacts with \(A\).
• Notice how each column of \(A\) in the provided solution is tailored to reflect the substantially aligned linear transformation of a single vector's image — embodied in vector \(\mathbf{w}\).

Matrix multiplication, while straightforward once understood, is a powerful tool that can encode complex transformations and system solutions.

Linear Algebra Concepts

Linear algebra is a field of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations. It provides the language and tools to express various mathematical and applied science concepts.
Key Concepts in Linear Algebra:

• Vector Spaces: A set of vectors with rules for vector addition and scalar multiplication, retaining structure under these operations.
• Linear Transformations: Functions between vector spaces that preserve vector addition and scalar multiplication.
• Basis and Dimension: A basis is a set of vectors that spans the vector space, and the dimension is the number of vectors in the basis.
• Eigenvalues and Eigenvectors: For a given linear transformation, these are scalars and vectors such that the transformation leaves direction unchanged.
• Rank: As mentioned, the rank provides the dimensionality of the vector space's image.

Understanding these concepts is crucial for interpreting the structure and functions of matrices in operations like those found in this exercise. They enable the systematic manipulation and transformation of spaces in both theoretical and applied contexts.