Linear Combinations, Span and Basis Vectors

• When you have a pair of numbers in a Vector, think of each coordinate as a scalar (ok? where is he going with this?)
• In the xy coordinate system there are two very special vectors.
  • The one pointing right with length 1: $\hat{i}$
  • The one pointing up with length 1: $\hat{j}$
• Think of our $x$ coordinate in the vector as a scalar that scales $\hat{i}$, and the $y$ coordinate as a scalar that scales $\hat{j}$ (oh that’s where he was going)
• In this sense, the vector that these coordinates describe is the sum of two scaled vectors (okay?)
• $\hat{i}$ and $\hat{j}$ (the unit vectors) have a special name btw. together they’re called the basis vectors
• We can pick any values for the basis vectors and have a perfectly reasonable coordinate system (from a certain point of view)
• also also any time u scale two vectors and add them like this, its called a linear combination of those two vectors

More terminology!! The set of all possible combinations you can reach with a linear combination of vectors is their span:

$a\vec{v}+b\vec{w}$ (where $a$ and $b$ vary over all real numbers, and $\vec{v}$ and $\vec{w}$ are our basis vectors)

• btw the span is everything unless your two basis vectors line up (then its just that one line)
• or zero if they’re both zero (unlucky ig)

A lot of times (because it gets crowded otherwise) people think of vectors as points (where the tip of the vector is the point) and the tail is at the origin. This scales up to 3D space too:

• With two vectors the span is usually a flat sheet tho

When you have multiple vectors and you can remove one without reducing the span, you can say those vectors are linearly dependent. But if $\vec{w}$ can’t be expressed in terms of $a\vec{v}$, they’re said to be linearly independent

(ch. 2)