[3Blue1Brown] Linear combinations, span, and basis vectors

-This post was written for arranging content of Essence of Linear Algebra series

   that uploaded by 3Blue1Brown channel in Youtube. -

 

If there's a certain vector $\overrightarrow{A}$ a start from Origin, in random plane in two dimension.

 

and that coordinate of $\overrightarrow{A}$ is the land loacation of $\overrightarrow{A}$.

in this case, it's $\begin{bmatrix}  3 \\ -2 \end{bmatrix}$

we need to think about each number of coordinate like scalar.

 

In the xy-coordinate system, like plane in two demension, there are two very special vectors.

the unit vectors pointing positive direction with length 1 in the x-direction and y-direction

we called each vectors to i-hat(x-direction) and j-hat(y-direction).

This vectors are the basis vector of x,y-coordinate system.

 

Back to the point, the coordinate of This vector $\overrightarrow{A}$ can explain as scalar for two basis vectors.

Let's scale the i-hat in 3 times, j-hat in 2 times and combine together.

 

the result of combining two basis vectors is the $\overrightarrow{A}$.

 

So then, are they i-hat and j-hat always basis vector for x,y coordinate system? 

NO, it's not. if you chose different basis vector, you got a completely reasonable new coordinate system.

 

If we can chose $\overrightarrow{v}$ and $\overrightarrow{w}$ for basis vectors,

then coordinate of that purple vector is $\begin{bmatrix}  -0.80 \\ 1.30 \end{bmatrix}$

And this coordinate is not same the case that the basis vectors are i-hat and j-hat $\begin{bmatrix}  3.1 \\ -2.9 \end{bmatrix}$,

 

One more thing, the set that can maded by Linear combination of $\overrightarrow{v}$ and $\overrightarrow{w}$ is the span of two vectors.

 

If there's vector x and can escape from the span of v and w,
In other words, when one or more vectors can be excluded without reducing the span, it is called linear dependency.

 

On the other hand, if the vector x can add another dimension to the existing span, it is called linear independence.

 

-The basis of a vector space is a set of linearly independent vectors that span the full space-