21.1: Basis Vectors

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Below is a really good review of concepts such as: Linear combinations, span, and basis vectors.

from IPython.display import YouTubeVideo
YouTubeVideo("k7RM-ot2NWY",width=640,height=360, cc_load_policy=True)

Question

What is the technical definition of a basis?

Question

Write three basis vectors that span \(R^3\).

From the above video two terms we want you to really understand Span and Linear Independent. Understanding these two will be really important when you think about basis. Make sure you watch the video and try to answer the following questions as best you can using your own words.

Question

Describe what it means for vectors to Span a space?

Question

What is the span of two vectors that point in the same direction?

Question

Can the following vectors span \(R^3\)? Why?

\((1,−2,3),(−2,4,−6),(0,6,4)\)

Question

Describe what it means for vectors to be Linearly Independent?

If you have vectors that span a space AND are Linearly Independent then these vectors form a Basis for that space.

Turns out you can create a matrix by using basis vectors as columns. This matrix can be used to change points from one basis representation to another.