Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when answering or asking about linear algebra basics like basis, span and linear independence, since it gives a geometric picture behind the definitions.
Recommended by 1 people across 1 different months, 2020 to 2021.
What it is
A 10-minute animated explainer from 3Blue1Brown, the second chapter of the Essence of linear algebra series. It builds up linear combinations, span, linear dependence and the definition of a basis using geometric pictures of 2D and 3D vectors.
Cited in 2 Wikipedia articles, including Basis (linear algebra), Linear span.
What people said
Recommends Nεo Pλατo · 2020-02 · Mathematics Stack Exchange · on “Linear Algebra - finding angle between two vectors”
If you are learning linear algebra, here's a gem: [LINK] Best linear algebra intro ever! Watch everything. But anyway, you can associate that to a transformation from $\vec w_1 \to \vec w_2$ . [this video] and [LINK] . Those explain how it works. Thus rotation becomes a mapping of where $\hat \imath, \hat \jmath$ go when defined by a matrix. One column for $\hat \imath, \hat \jmath$ each. But in your case $\hat \imath=\vec w_1$ so I'll just approach it directly. Both basis vectors are on a unit circle. So any point on that circle is defined by the co-ordinates $(\cos(\theta), \sin(\theta)) \qquad \text{Check the diagram.}$ [LINK] So if $\hat \imath$ has been rotated, it has moved 120 degrees alon
Cites harry · 2020-10 · Mathematics Stack Exchange · on “Linear independence in 3D vector space using 2 vectors”
According to this [this video] video, two vectors can't define more than one plane (from around 5:58 to 7:23), and it takes three vectors to go beyond that one plane. Wouldn't two vectors be enough to define the span of all vectors in 3D space? Why would we need three?
Cites Quazi Irfan · 2021-08 · Mathematics Stack Exchange · on “Why it is not enough to define a basis as a set of linearly independent vectors?”
I am going through 3b1b linear algebra video, and at the end of 2nd video [this video] in the playlist [LINK] , he defines basis as The basis of vector space is a set of linearly independent vectors that span the full space. My question is, isn't it redundant to say the part that span the full space because a set of linearly independent vector always span a subspace or a space. For example, if we have two 3x1 independent vectors then their span a 2d subspace in a 3d vector space. So, why it is not sufficient just to define a basis as a set of linearly independent vectors?