Oct 25, 2022 · Actually, row vector and column vector aren't the same thing. As a beginner think of 1x3 matrix as three one-dimensional vectors and 3x1 matrix as one three-dimensional vector.

Indeed, it's often just aesthetics; given a vector space of column vectors, it's isomorphic to the vector space of row vectors (the isomorphism being the transpose map). The distinction is …

Jul 13, 2022 · A vector is just an element of a vector space. A column vector is an element of the vector space of all column vectors. So in that sense, all column vectors are vectors. As for …

By the way, if one works entirely in terms of matrices, and considers any vectors to be a column matrix, then the inner product can be defined by $\mathbf {v} \cdot \mathbf {w} = \mathbf …

Jul 27, 2022 · A column vector is a true vector, whereas a row vector is something that eats a vector and spits out a scalar. But maybe you think we can just "flip" a row vector to obtain a …

Nov 2, 2021 · If one thinks of tangent vectors as column vectors, then $\nabla f$ ought to be a column vector, but the linear functional $\langle -, \nabla f \rangle$ ought to be a row vector. A …

Jun 26, 2016 · I would also point out that rows and column vectors are not the only choices you have. For example the space of two by two matrices with coefficients in a field form a vector …

Mar 19, 2020 · The question is whether you can take the dot product of a column vector and a row vector.

Apr 23, 2020 · The abstract answer is: a vector is a member of a vector space. A vector space is any set that has two operations (addition and scalar multiplication) satisfying certain …

Dec 15, 2017 · Multiplying column or row vectors are simply special cases of matrices in general, so that condition still applies. In short: it's a consequence of the (usual) definition of the …