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I am looking at the 3Blue1Brown vector transformation video and at 2:17 he gives an example of a transformation that rotates space 90 degrees over the y-axis.

Based on the intuition I have it seems weird because he says that $\hat{i}, \hat{j}$ and $\hat{k}$ are all basis vectors. Therefore in a 3-dimensional Euclidean space the vectors should have the values $(1,0,0), (0,1,0)$ and $(0,0,1)$ respectively.

Now I have the understanding that every transformation can be expressed as some scalar multiplicaton of the basis vectors. For exaple if I have a vector $\overrightarrow{v} = (2,3)$ where the $2$ and $3$ are the coordinates of the vectors that can be represented in relation to the basis vectors: $\overrightarrow{v} = (2\hat{i}, 3\hat{j})$ on a two dimensional plane and want to rotate it 180 degrees then the transformation could also be expressed as multiplying each coordinate of the vector with a scalar: $(-1\cdot \overrightarrow{v}, -1\cdot \overrightarrow{v})$ or using matrices $\begin{pmatrix}2\\ 3\end{pmatrix} \begin{pmatrix} -1, -1 \end{pmatrix}$.

That said in the video where he performs the transformation visually he shows how the $\hat{i}$ changes from the x-axis to the z-axis which is weird because to perform some transformation changing axies should not be neccessary and in my intuition cannot be done because we have defined $\hat{i} = (1,0,0)$

My question is that if we assign concrete values for for example $\hat{i}$ then we shouldnt be able to transform them from one axis to the other, all basis vectors represent one axis but in the video the basis vectors change axies

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Let's consider $2$ dimensions first. When we perform a rotation of the plane by $90º$ counterclockwise, what we're really doing is moving $\hat{i}$ to $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ and $\hat{j}$ to $\begin{pmatrix} -1 \\ 0\end{pmatrix}$. In other words, $\hat{i}$ is now pointing in the positive $y$- direction, and $\hat{j}$ is now pointing in the negative $x$-direction.

Because addition is preserved under a linear transformation (transforming the sum of two vectors is the same as transforming each separately then adding them together), and because scalar multiplications are preserved ($f(u + v) = f(u) + f(v)$ and $f(cu) = cf(u)$ more formally), we can decompose every transformation in terms of these basis vectors.

So in the case of $3$ dimensions, or $n$ dimensions, we can move the basis vectors anywhere, even to another axis, and any transformation can be thought of in this way.

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