Revision Notes : Essence of Linear Algebra

7 min readAug 12, 2019

Video Playlist Followed :Link to the Video by 3Blue1Brown

Video 1 :

Meaning of Vector for Physicist (arrows pointing in space), CS Student (ordered list of numbers, sets), Mathematician ()
In Linear Algebra the Vector will always origin at 0,0 in Space
Vectors are coordinates as to how much distance along the X-axis or Y axis , that can also be represented in form of a 1X2 Matrix
(Addition)Sum of Two Vectors = move along vector 1 , then move along vector 2 then the total travel is the sum of the two vectors (ie Matrix addition )then following the end point from the origin
(Scaling)Multiplication of vectors : scaling the factors by a factor multiplied

i vector is x-axix and j is y-axis
i and j are the Basis of the Coordinate System (and everything else is just scaling these basis in the coordinate system)
[Technical Definition :] The Basis of a vector is a set of linearly independent vectors that span the full space
The choice of basis vector matters when choosing a coordinate system
Span : of two vectors is set of all linear combinations (ie the points they can reach)
If adding two vectors keeps the resultant vector span in the same dimension then they are Linearly dependent , and if their addition adds a new span of dimensions then they are “Linear independent”

Linear Transformation :Linear (all lines must remain lines and Origin remains fixed )+ Transformation (a function that takes an input and gives an output, the transformation is used instead of function to bring int the role of moment of vector from its initial position to final position )

The Place where original vector lands after the transformation is used as value to be multiplied with other vectors to find where they land after the transformation

If you want to apply linear Transformation many(Rotation +Shear) times than that is called the Composition

The Determinant of Matrix is the change in the area covered by the Vectors after the Transformation

It can be Visually Interpreted as the Transformation required so that the Vector A lands on vector V after the Transformation
Inverse Transformation is the transformation required from V to go back to A (AInv A is Identity since it ends up doing nothing )
Unless the the Determinant is Not ZERO the Determinant will Exist (if zero the area will be line and you cannot decompose back line back to area or volume through a single function)
A Inverse X A is doing Nothing thus the Identity matrix
Solution can Still exist if the Determinant is zero if the solution exist on the Same Line
When the output of Determinant is 0 (it squeeze down to a line ) then the Rank of the Matrix is said to be 1 , if the Transformation lands on a 2-D Plane instead of a line than it has the Rank 2
RANK : is the number of Dimensions in the Output
Be it Line of Plane : it is called the Columns Space of a Matrix , and column tells where the base vector lands
Span of Columns = Column Space

0,0 is always in the Column Space since in the Linear Transformation the origin cannot be moved and any number of vectors can land on the origin after the Transformation thus it is the Null Space (Kernel of Matrix)of the Vector

Dot Product is projecting a Vector onto other vector and Multiplying their Length , if the projection is in other direction the Dot Product will be Negative , and if Perpendicular then the projection will be zero

Duality : Natural correspondence (ie linear transformation of vector is some other vector in that space ) (ie 2 computation that look similar )

take a vector v and move the vector w to the end of it and then repeat the same in other order (ie take w and move start of v to end of w)the diagram obtained is a parallelogram ) — the Determinant
+ve and -ve is due to the order (j should be on left of I anticlockwise )

The Cross product is not a number its a resultant vector as per Right hand thumb rule of the magnitude given by the number

getting to the same vectors but by using different basis means scaling differently to land and the same coordinates (describing same thing but in different languages )
drawing different grid lines for same vectors

when a vector is transformed the span of it also changes but sometimes even after transformation the span doesn't changes
For those vectors which remain on their span even after transformation but stretched are called the eigen vectors of the transformation and eigen values are the factors by which they are stretched or squished