What is the rank of a vector?
From linear algebra we know that the rank of a matrix is the maximal number of linearly independent columns or rows in a matrix. So, for a matrix, the rank can be determined by simple row reduction, determinant, etc. However, I am wondering how the concept of a rank applies to a single vector, i.e., $\mathbf{v} = [a, \ b, \ c]^{\top}$. My intuition suggests that the rank must be equal to 1, but I'm not even sure if it is defined for a vector. Can anyone help shed some light on this issue?
Thanks in advance.
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$\begingroup$A vector is an element in a vector space. As such, it has no rank. A matrix, in the context of linear algebra, is interesting because it represents a linear transformation between vector spaces. It's the linear transformation we care about, not the rectangle of numbers we call a matrix. A linear transformation has a rank and that rank is the dimension of the image of the linear transformation. It's an interesting concept since it's a measurement of how large the linear transformation is. It just so happens that when the linear transformation is represented as a matrix the rank can be computed in various convenient ways.
With this in mind, the superficial resemblance of a vector to a matrix is misleading. A vector is not a representation of a linear transformation between vector spaces. For that reason, it simply makes no sense to ask what the rank of a vector is.
$\endgroup$ 22 $\begingroup$Another definition of rank of a matrix is the dimension of the vector space spanned by its column, so any non-zero vector spans a space of dimension 1.
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