Brushing Up Linear Algebra-Vectors
This is my first blog post of my Blogging Series “Brushing Up Linear Algebra”
Most of us try to learn Mathematics and its applications but end up just learning the numerical computations.Mathematics is more than numerical computations.The Magic of Mathematics begins when we understand the geometrical interpretations of the numerical computations.Linear Algebra is not just about Matrices,Determinants and Vectors its about Understanding.
This post is about Vectors.
Almost each of us would have been come across vectors whether we study Mathematics,Physics or Computer Science.
Lets try to understand Vectors from each of the perspectives-Mathematics,Physics and Computer Science and build an intuition about Vectors.
Physics :Vectors are arrows pointing in space.What defines a vector is its length(magnitude) and direction.As shown in image the tail can also be called has Initial point and the head as Terminal point
Computer Science : Vectors are ordered list of numbers or we can say another name for list.You would be more familiar with this definition if you have studied R.
Lets us say we are defining a car on the basis of its kmph and its Price.Then we can represent the car using a vector say [35kmph , Rs10,00,000] .
Mathematics : Vectors in 2 Dimensions are geometrically represented by a directed line segment between two points in space.
Lets combine all the above definitions and understand it more better.Imagine a coordinate space such that the initial point of the vector lies on the origin (0,0) always and tail of the vector lies at any point in coordinate plane.
Referring the image,tail lies at the point (2,3).Which means in x axis it spans a distance of 2 units and in y axis it spans a distance of 3 units.
Thus I can say that while considering the definition from Physics(Arrows pointing in space with initial and terminal points),if I keep the initial point on the origin always I reach to the mathematical definition of Vectors(directed line segment between two points in space -one being the origin and other being the terminal point).
Put the Terminal point in a list say [2,3] and we reach to the computer science definition.Thus,
I hope now you can imagine vectors more easily.
Before moving ahead,few key points :
Vector with no length is called zero vector.
Negative vector is just same as positive vector but in opposite direction.
Vector Addition :
Physics :
Mathematics :
Computer Science :
If we have two vectors [x1,y1] and [x2,y2] ,addition of these two vectors would be the addition of respective terms i.e [x1+x2 , y1+y2] which can also be visualized in the above diagram shown for mathematics.
Scaling of Vectors :
Multipying a vector with a scalar(s),scales the vector s times.
Linear Algebra revolves around vector addition and Scalar Multiplication.I hope after reading this post you have a better understanding of geometric implementation of vectors.
Happy Learning!!
