# Linear Transformations

### Rethinking

One of the best, earliest insights that I got from watching 3blue1brown’s excellent videos on The Essence of Linear Algebra is how to conceptualize a linear transformation performed by a matrix, `A`

.

For starters, he conceptualizes everything he talks about relative to the unit vectors `i`

and `j`

, which are vectors of magnitude `1`

that point in the `x`

and `y`

directions, respectively.

```
from IPython.display import Image
Image('images/basis_unit_vectors.PNG')
```

Thus, for any matrix, `A`

, that defines a linear transformation, we should make it a habit to immediately identify the columns of `A`

as mappings for these unit vectors.

That’s to say that for a matrix that looks like

```
a b
c d
```

Then the first column, `[a c]^T`

, corresponds to where the `i`

vector winds up. Similarly, `[b d]^T`

is where `j`

does.

Verifying this computationally is trivial (and a little tedious), but internalizing the relationship between columns of `A`

and unit vectors makes framing *what’s actually happening* in the transformation a breeze.

### Composition

He expands on this visual intuition in his video on compositions of multiple linear transformations, encouraging you to think of the product of two matricies `M3 = (M2)(M1)`

as the same as “applying `M1`

, then applying `M2`

” but in one step.

Moreover, we can lean on our intuitive, “first column of `M1`

is where `i`

goes”, and think of the `M2`

, `M1`

multiplication step in two parts:

`Image('images/composition_i.PNG')`

## Preserves Spacing

Finally, he iterates again and again throughout his series that a nice consequence of a linear transformation is that all parallel and evenly spaced points/gridlines in our original space *continue being parallel and evenly spaced after our linear transformation*. In this image, the gray lines represent our typical `(x, y)`

space, and the blue and white lines represent our transformation to a new space using the matrix

```
1 1
0 2
```

`Image('images/gridlines.PNG')`

this property comes up a ton in later sections