Span

3Blue1Brown notes for Math 232

Span of the coordinates

• The hats

  • we can think of the x and y axis as these scalars $\hat{i}$ and $\hat{j}$

    • it is like the complex plane (WOW)

      • IE:

$$\begin{array}{c}3\\ -2\end{array}=(3)\hat{i}+(-2)\hat{j}$$

• Very important → “basis vectors” - showing the xy coordinate system

  • We can make our own basis vectors (hats)

    • to make infinite ways to make one vector

• linear combination

  • when we add some type of basis vectors it creates a linear combination

  • the “span” of $\vec{v}$ and $\vec{w}$ is the set of all their linear combinations

    • $a\vec{v}+b\vec{w}$
      • letting a and b very over all real numbers

        • This allows us to write any vector on the 2d plane

• Span in 3d space

  • 2d Span in 3d space is like 3d printing

    • when we try to take the span for a 2d vector (vector with two numbers only) in 3d all the possible values (span) will only draw a flat 2d plane through the origin of teh 3d space

  • 3d span in 3d space

    • Again the span for this set is all possible linear combinations

      • $a\vec{v}+b\vec{w}+c\vec{u}$
        • with linear combinations for $\vec{v},\vec{w},\vec{u}$

      • however if the third vector is “trapped” in another vector it will produce the same span as a 2d vector intuitively

Linear dependency

• Redundent vector

  • When a vector is trapped within another vector it is called

    • Linearly dependent

• You can write the linear dependent vector as a linear combination of the other vectors

  • $\vec{u}=a\vec{v}+b\vec{w}$
    • for some values of $\vec{a}$ and $\vec{b}$

Linear independence

• Non reduntant vector

  • When teh vector is actually contributing it is called

    • Linear independent

• You can describe this

  • $\vec{u}=/=a\vec{v}+b\vec{w}$
    • for all values of a and b