Is Everything inside a matrix a vector

Question

Lets say we have a matrix $$\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$ Is this matrix also a vector? If yes then what does this matrix mean, I mean what vector will it form??

Answer 1

Because there are different definitions of what constitutes a "vector" in different contexts, this is a question that does not have a simple yet totally correct answer. However, in the context of introductory linear algebra or physics that isn't tensor-related, we can say this:

No, a matrix is not a vector. A vector (or "column-vector") specifically refers to a vertical list of numbers. So, the "matrices" $$ \pmatrix{1\\1}, \quad \pmatrix{1\\1\\1\\1}, \quad \pmatrix{1\\2\\-1\\0}, \quad \pmatrix{0\\1\\0}, $$ are all vectors. For convenience, these vectors are often written horizontally. For example, the third vector above might be written as $(1,2,-1,0)$ or $(1,2,-1,0)^\top$, depending on which textbook you're using.

"Geometrically", the vector $(1,2,-1,0)$ would correspond to the arrow in $4$-dimensional space that goes from the coordinates $(0,0,0,0)$ to the coordinates $(1,2,-1,0)$.

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In the modern and more general mathematical definition of a vector, a vector refers to any object of a "vector space," and a "vector space" is any group of objects with a sensible notion of addition and multiplication by scalars. So, for example, the set of $2 \times 2$ matrices is considered a "vector space" because these operations make sense. For example, $$ -2 \cdot \pmatrix{1&2\\-1&0} = \pmatrix{2&4\\-2&0}, \quad \pmatrix{1&2\\-1&0} + \pmatrix{1&1\\1&1} = \pmatrix{2&3\\0&1}. $$ In this context, the "geometric" interpretation wherein a vector has "magnitude and direction" no longer applies. A matrix, while it is a "vector" by this other definition, does not have magnitude and direction.