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The outer product is an operation between two vectors resulting in a matrix. When you have two vectors, one denoted as \( \mathbf{u} \) in \( \mathbb{R}^{m} \) and the other as \( \mathbf{v} \) in \( \mathbb{R}^{n} \), their outer product is calculated by multiplying each element of \( \mathbf{u} \) by every element of \( \mathbf{v} \). This creates an \( m \times n \) matrix.
- Think of it as creating a grid where each cell contains the product of one element from \( \mathbf{u} \) and one from \( \mathbf{v} \).
- If \( \mathbf{u} = [u_1, u_2, \, \ldots \, , u_m]^T \) and \( \mathbf{v} = [v_1, v_2, \, \ldots \, , v_n] \), the resulting matrix \( A = \mathbf{u}\mathbf{v}^T \) has elements \( a_{ij} = u_i v_j \).
This matrix has a very special property — every column is a scalar multiple of \( \mathbf{u} \). As a result, all column vectors point in the same direction in \( \mathbb{R}^m \), which is a critical characteristic when determining the rank of the matrix.

The column space of a matrix is an important concept in linear algebra. It refers to the set of all possible linear combinations of the column vectors of a matrix. This space tells us about the span of the matrix columns in a multidimensional space.
- For instance, if you visualize the columns of an \( m \times n \) matrix \( A \), think of each column vector as an arrow starting from the origin stretching into space.
- The column space is the collection of all endpoints that you can reach by combining these arrows according to the scalars of the combination.
When a matrix has rank 1, its column space collapses into a single line, showing all columns are essentially stretched versions of a common vector \( \mathbf{u} \). This means the matrix doesn't explore much of the space — just one line through the origin.

Linear dependence refers to a situation where some vectors in a vector space can be written as linear combinations of others. When understanding matrices, this concept helps us analyze their columns or rows.
- A set of vectors is linearly dependent if at least one vector can be described as a combination of the others.
- For rank 1 matrices, all vectors are linearly dependent.
More specifically, in a rank 1 matrix, all columns are multiples of each other. Thus, they are not spanning a larger dimensional space but are confined to the smallest possible space — a single line as mentioned earlier. This is what makes them linearly dependent and indicates a reduction in the diversity of directions the matrix can project onto.

The proof that a matrix has rank 1 if and only if it can be expressed as an outer product \( \mathbf{u}\mathbf{v}^T \) relies on understanding both necessity and sufficiency.
- **Necessity:** If a matrix \( A \) is rank 1, all its columns are scalar multiples of one column vector, \( \mathbf{u} \). This means \( A \) can be expressed as an outer product \( \mathbf{u}\mathbf{v}^T \), where \( \mathbf{v} \) compiles the necessary scalars.
- **Sufficiency:** Conversely, if \( A = \mathbf{u}\mathbf{v}^T \), it inherently has rank 1 because each column of the resulting matrix is a multiple of \( \mathbf{u} \), ensuring all columns align consistently in the same direction.
This dual aspect shows that writing a matrix as an outer product is both necessary and sufficient for it to have rank 1, affording a complete picture of its structural simplicity.