Tutor profile: Shubham K.
Subject: Linear Algebra
What is the definition of a vector space?
A vector space V is considered to be a set of vectors over which two operators are defined: the scalar multiplication, and the vector addition. Under vector addition, you can take two vectors belonging to the vector space and simply add each of their corresponding elements together to get a third vector (which will also belong to V). Under the scalar multiplication, you can multiply a vector with a number/scalar to produce another vector (which will again belong to V). No other operations are allowed on the vector space, by definition. Based on just these two operators, we can define 10 fundamental properties on the vector space (for any number of vectors A, B, C and so on belonging to the said vector space): 1) Closure under vector addition: For A, B belonging to V, A+B also belongs to V 2) Commutativity of vector addition: A+B = B+A 3) Associativity of vector addition: A+(B+C) = (A+B)+C 4) Existence of Additive Identity: There exists a vector belonging to V such that any other vector added to it would give back the same vector: A + 0 = A; then 0 (the additive identity) belongs to V. 5) Existence of Additive Inverse: There exists a vector belonging to V such that when the additive inverse of this vector is added to it, you get the additive identity vector: A + (-A) = 0; then, here, (-A) is the additive inverse of A. 6) Closure under scalar multiplication: For a vector A belonging to V, any scalar multiplication of A (for eg. 2A, -3A, (5/7)A) also belongs to V 7) Distributive Property for scalars: If we have an expression where a scalar is multiplied to a vector addition of two vectors belonging to V, we can simply distribute the scalar amongst the two vectors and then apply vector addition: for eg. 2(A+B) = 2A+2B 8) Distributive Property for vectors: If we have an expression where a vector is scalar multiplied to the sum of two scalars (just addition of two real numbers), then we can distribute the vector amongst those scalars: for eg. (2+3)A = 2A + 3A 9) Associativity of scalar multiplication: for some scalars a, b, it is true that a(bA) = (ab)A 10) Existence of Scalar Identity: There exists a scalar such that scalar multiplication of any vector with this scalar would give the same vector: 1(A) = A; 1 is the Scalar Identity. And just with these intuitive properties, which can be easily verified by just keeping scalar multiplication and vector addition in mind, we have defined a vector space. Concepts of linear dependence/independence, basis, dimensionality, and so on will follow after this.
I am a 12th-grade student. Why should I care about studying calculus?
Well, at first glance, to many, the subject might seem to be a mathematician's best friend and a 12th grader's worst enemy. I understand how from a 12th grader's perspective the subject might seem quite overwhelming, with the integral sign, and the limits, but at the end of the day, it is nothing but a toolkit, which helps you in calculating areas under curves; it doesn't bite. And while that above "definition" might be too boring for you, let me also remind you that Calculus has enabled humans to take a giant leap without which many daily necessities like the internet or even your mobile phone would only be a dream; it also enabled us to go to the moon! It is exciting, intuitive, and one of the most important mathematical tools that we possess today.
What is the most important part of Physics which every student pursuing a course in the subject should always keep in mind?
There are debatably a lot of very important topics in Physics, ranging all the way from science of the smallest, that is, quantum mechanics, to the science of the biggest, that is Cosmology or Astrophysics. I don't think I would ever be in the position to rank these topics one over the other; people have devoted entire lives trying to unravel little mysteries about these different topics. However, the most important part of Physics is not any one of these topics, to begin with; the gist of the subject comes from the fact that it can be verified in nature. Thus, the most important part of Physics is a step beyond these concepts and topics, it's about going out in nature and actually finding what these topics predict. The entire foundation of the subject is based on the fact that nature acts in a certain way, and thus by definition, this subject has to be able to ascertain exactly how nature does that.
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