Hour 3 & 4 - Ranks, Solutions, Vectors & Linear Combinations and Span

Hello everyone! Welcome to the second blog! Last time, we learnt about linear systems and how to solve them. Now, we are going to learn about ranks, vectors, linear combinations and span.

Hour 3

Rank

A rank of a matrix is the number of “leading ones” in the coefficient matrix.

Now, why is a rank of a matrix important? By finding the rank of an augmented matrix,(the number of leading ones in the rref of the augmented matrix) we can actually find out how many solutions our system has!

Unique Solution

Here, as the matrix’s columns have all leading ones, it must have solutions equal to the number of columns of the matrix. All variables are leading and each has one particular solution!

Infinitely Many Solutions & Consistency

In any consistent system, keeping aside the leading variables, if there is atleast one free variable, then the system will have infinitely many solutions!

However, if the augmented part has any real number other than 0, and the matrix row has all 0’s, the system will be inconsistent as 0(x) + 0(y) cannot be equal to anything other than 0.

Vectors

A vector is a matrix which has only one column and n rows, where n is a positive integer. Thus, a vector has dimensions n x 1. Some operations with vectors are shown below:

Some helpful links

Tutorials->

(1) https://youtu.be/MxGJeli6qOc

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

Exercises->

(1) https://www.math-exercises.com/matrices/rank-of-a-matrix

Hour 4

Linear Combinations and Span

Now that we have learnt what vectors are, let’s see what a linear combination of vectors can look like:

Here, we can see that the solution vector could be any coordinate along the line y=x. We can say that for the vector (1,1), it’s scalar multiple is t(1,1), where t is any real number.

Linear Combination

A linear combination of the vectors v1, v2,…, vn is an expression of the form c1v1 + c2v2 + ··· + cnvn where c1, c2, through cn are real numbers. So it’s just a sum of multiples of the vectors v1, v2,…, vn.

Span

The span of the vectors v1,v2,…,vn is all possible linear combinations of these vectors, and it is denoted by span(v1,v2,…,vn).

As you guys can see, it has a solution for (1,1) and I’d like you guys to try out whether there is a solution for any coordinate in the plane!

Some helpful links

Tutorials ->

(1) https://www.youtube.com/watch?v=k7RM-ot2NWY

(2) https://www.youtube.com/watch?v=ivP-6oicIWU

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