(a) Prove that every 2 ×2 matrix of rank 1 can be written in the form A=𝐮 𝐯^T where 𝐮, 𝐯 ∈ℝ^2 ar

Step 1: Understand the rank of a matrix.u000AA $2 u005Ctimes 2$ matrix $A$ has rank 1 if and only if it

(a) Prove that every 2 ×2 matrix of rank 1 can be written in...

(a) Prove that every $2 \times 2$ matrix of rank 1 can be written in the form $A=\mathbf{u} \mathbf{v}^T$ where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^2$ are non-zero column vectors. (b) Which rank one matrices correspond to orthogonal projection onto a one-dimensional subspace of $\mathbb{R}^2$ ?

Key Concepts

Rank of a Matrix

The rank of a matrix is the dimension of the vector space generated by its columns (or rows), which reflects the number of linearly independent columns. For a matrix of rank 1, all columns are scalar multiples of a single vector, implying that the matrix has the simplest nontrivial structure. This concept is central to understanding how such matrices can be decomposed and manipulated, as it directly informs the factorization into products of vectors.

Outer Product Factorization

Outer product factorization is the process of expressing a matrix as the product of a column vector and a row vector, written as u v^T. In the case of rank one matrices, this factorization is always possible and unique up to scalar multiples, because every entry of the matrix can be generated by the product of the corresponding entries of the two vectors. This concept is fundamental in linear algebra as it simplifies many problems by reducing a matrix to a product of lower-dimensional objects.

Projection Matrices

Projection matrices are used to represent the operation of projecting vectors onto a subspace. An orthogonal projection matrix, in particular, satisfies the properties of idempotence (A^2 = A) and symmetry (A^T = A). When projecting onto a one-dimensional subspace, the projection matrix is of rank one and can be written as the outer product of a unit vector with itself. This characterization is critical in both theoretical linear algebra and practical applications such as numerical analysis and data projection.

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