# Finding the change in basis vectors from a linear transformation

At about $4:00$ my question pertaining to 3Blue1Brown's video of a collection of linear algebra questions begins. The video is here.

• After a linear transformation, the vector changes, but so do the "boxes" in the "graph" of the vector space, if that makes sense. Before the transformation, the boxes in the vector space looked pretty standard, and then after the transformation, they seemed to increase in scale. If you keep your eye on the box where the tail of the vector $-1 \hat i + 2\hat j$ lands, you can see it's stretched and rotated to be far larger. Thus, there is a new "graph" or coordinate plane we're working in. But why then, does the narrator classify the new basis vectors in terms of the dulled out, original coordinate plane, with the original, standard-oriented boxes, calling transformed $\hat i$ $\hat i -2 \hat j$ when it's still a basis vector, so it shouldn't be a $\hat j$ in it to begin with? It should just be the new $\hat i$ with $1$ as its length in $x$ and $0$ as its length in $y$.

• Also, all of these kinds of transformations imply that our vectors start in the identity matrix form, so I'm assuming something like a "Shear" can only be classified if our $\hat i$ and $\hat j$ matrices are in the traditional orientation.

• It might be useful to provide a link to the video. Also, don't learn linear algebra from the videos, but use the videos once you already have read a book about it.
– user370967
Aug 22 '17 at 20:56
• @Math_QED Goodness me, bad mistake. And why do you think that's reading about it first instead of a video introduction is better? Aug 22 '17 at 20:59
• He explains everything quite well, but simplifies a lot of things. E.g. in this matrix video, he always starts with the canonical base of $\mathbb{R^2}$ and then explains how the matrix of the transformation will look like. However, things get more complicated when one starts in an other basis of the vector space. Also, the videos only handle about geometrical spaces, while linear algebra also treats spaces of functions etc.
– user370967
Aug 22 '17 at 21:02
• @sangstar The only way to really learn anything in mathematics is to apply and test your understanding with an exercise, the kind that you'd find in a textbook. It's usually easier to fill in any gaps in your understanding with a textbook than it is with a video. Aug 22 '17 at 21:53

I think you're getting confused as to what exactly a linear transformation does. Remember that $T$ takes a vector (either a point with respect to the objective, "dull" grid or a pair of coordinates) and produces an output vector. So in the context of the video, $T(\hat i) = \hat i - 2\hat j$. Note that $\hat i$ refers to the specific vector $(1,0)$ in 2-dimensional space; it does not simply mean "the first vector of our basis". While it is true that the vectors $T(\hat i)$ and $T(\hat j)$ form a new basis (for the particular transformation illustrated), those vectors are different from the inputs $\hat i$ and $\hat j$. It does not make sense to say that $T(\hat i)$ "is the new $\hat i$".