Hour 5 & 6 - Spans and Vectors & Linear Independence, Basis and Dimensions

Hey everyone! Welcome to our third blog. In the last blog, we covered vectors, linear combinations and spans. Let’s dive a little deeper now!

Hour 5

Span

A span of vectors is defined as all linear combinations of the vectors.

Here, as u2 is a multiple of u1 (u2 = -2u1), u2 itself is a linear combination of u1. Hence, the span(u1,u2) = span(u1).

Geometric Respresentations

2D

Here, the paralellogram represents the area covered by the span of the vectors where s and t can be any value between 1 and -1.

3D

Two non-colinear vectors in R3 will span a plane in R3.

Strategy to Visualise

In order to visualise the span of vectors and what shape the span might be, you guys can follow the following strategy:

Hour 6

Linear Dependency

Now that we learnt about linear combinations of vectors, let’s learn about linear independence. A set of vectors are linearly independent if they have a trivial relation. This means that each vector in the set of vectors must be multiplied by 0 in order for their sum to be 0.

Linearly Dependent

A set of vectors are linearly dependent if they have a non-trivial relation.

Linearly Independent

A set of vectors are linearly independent if they have a trivial relation.

If two vectors are linearly independent, their augmented matrix will have the same number of leading ones as the column of the matrix.

Basis and Dimensions

Basis: v1,v2,…vn are a basis of a subspace V if they span V and are linearly independent. The dimension of the subspace V is the number of vectors in the basis of V. If there is no subspace, no dimensions exist.

Some helpful links

Practice!

Tutorial

(1) https://youtu.be/yLi8RxqfowA

(2) https://youtu.be/X3C4ATIO3bo

Exercises

(1) https://www.math.colostate.edu/~aristoff//369_HW15.pdf

(2) https://onlinemschool.com/math/library/vector/linear-independence/

That’s all for today! See you in the next blog :’)