# 3blue1brown and the visual argument that a vector is fundamentally different from a matrix

After watching the wonderful playlist of Linear Algebra on the YouTube channel 3blue1brown, I realized that vectors and matrices are, fundamentally, different concepts:

• Vectors are numerical entities in a $$n$$-dimensional space.
• Matrices are how theses numbers are (linearly) transformed.

I am restricting it to a Linear Algebra. Of course, for multilinear algebra, matrices (and tensors) are, in fact, numerical entities in the same way that a vector is. Regardless, for Linear Algebra, this argument seems to have a much deeper interpretation that what is commonly said about vectors and matrices:

A vector is just a one-column matrix

I am not saying that such answer is wrong, once denoting a vector as a column is merely a convention. But doesn't it seem that this argument leaves out the main purpose of using matrices in linear algebra, which is apply linear transformations on vectors though matrix product,i.e., $$\mathbf{Ax}$$?

• You can think of a vector in $V$ as a linear transformation $\mathbb{R}\to V$. This aligns with both thinking of matrices as linear transformations and vectors as 1 column matrices. I would disagree that vectors and matrices are "fundamentally different" Dec 11, 2022 at 23:32
• Adding to what IsAdisplayName said, often in maths the vector space you are working over is a set of matrices (for example in Lie algebra) so there isn't really any clear difference between the two concepts beyond what vector space you are concerned with. Dec 11, 2022 at 23:58

A matrix is a concrete instance of a linear transformation when you've chosen a specific basis. However, when you move from linear algebra over the field of Real Numbers to an abstract field $$F$$ it becomes more helpful to view linear algebra more abstractly as linear transformations of vectors instead rectangular grids of numbers.
If $$V$$ is a vector space over ($$F$$) and $$a \in V$$, then a linear transformation $$T$$ is a function from vector space $$V$$ to some other (possibly identical) vector space $$W$$ (over $$F$$) such that $$T(ca)=cT(a), T(a+b) = T(a) + T(b)\;\;\forall a,b \in V, c \in F$$.
In this viewpoint, the difference between the vector and its transformation becomes clear. However, in the matrix viewpoint, you can have vector space of, say, $$2\times 2$$ symmetric matrices and a $$2\times 1$$ column vector being the linear transformation, which will be a transformation from $$S_{2,2}$$ to $$\mathbb R^2$$, in this case the roles are flipped.
Note: as pointed out by ex.nihil, if we want to keep the usual format of $$Tv$$ for a linear transformation $$T$$ applied to vector $$v$$ then we’d use the $$1 \times 2$$ row vector as the transformation $$T$$ applied to the matrix as $$v$$.
• In your last example, I believe you mean that the $2×1$ vector is acting as a right-multiplication transform; is this true? Because otherwise—when using the more typical left-multiplication for transforms, your example should instead be a $1×2$ vector that left-multiplies the matrix to map it from $S_{2,2}$ into $\mathbb{R}^2$. I know this distinction may be pedantic, but I guess it clarifies notational confusion. Please correct me if I'm wrong! Aug 19, 2023 at 16:33