IIT Madras MS Interview Experience

Primary Subject :- I have chosen Linear Algebra as my primary subject.

	There are 6 faculty members present on Interview Panel, One of the faculty member start with Linear Algebra. lets called him I1.
I1 :	Which topics you have covered in Linear Algebra ?
Me :	Vector space , Basis , Eigen Values and Eigen Vectors , System of Linear Equations , Matrix Decomposition.

I1 :	Let's start with Vector space. What is the Vector Space ?
Me :	Explained all 7-8 axioms that must be satisfied by a non-empty set of elements to be qualified as a Vector space. [ You will find the detailed answer here IISC Video 40:00 onwards ]

I1 :	What is the subspace of a Vector Space V ?
Me :	It is a non-empty subset of Set V such that the subset follows the Closure property " under the addition and under the Scalar Multiplication ". As V is already Vector space we have to check only above 2 properties for a subset of V to called it a Subspace of Vector space V. the subset must be non-empty because we always have zero element as a part of the subspace of any vector space V as we can always get zero element by scalar multiplication of any vector by scalar value 0.

I1 :	Now suppose there is a line described by x + 2y = 3 , can we say that it forms a vector space ?
Me :	No, as zero element (i.e (0,0)) must be part of any Vector space but here zero element does not satisfy the equation hence not part of vector space so this line does not form a vector space.

I1 :	Can we have " n*n matrix A " for which Column space = Null space ?

	[ I was confused at this question and try diff examples for the next 1 min but didn't get any such matrix. ]

I1 :	Try for 2*2 first.
Me :	Again after trying diff examples i able to make one such example [1 -1] [1 -1] [ Note : I am able to find this example only because i have seen the Gilbert Strang video on Matrix-Vector multiplication using 3 diff ways, i used the Column View of Matrix multiplication where he said that " vector that you got after A*x is the Linear Combinations of columns of A " You can find that lec here ]

I1 :	Now what can you say about this kind of n*n matrix A?
Me :	After thinking for 30-40 sec i told that as we know that rank of A = dimension of col space of A nullity of A = dimension of null space of A but for a matrix A we have col space = null space => both have same dimensions => rank = nullity also rank + nullity = n so we got rank = nulity = n/2 also matirx A must have even dimension i.e n = even number ( i.e : A can be 22 , 44, ...)

I1 :	Can you give me more generalize property ?
Me :	i tried some examples but couldn't get anything new.

I1 :	He told me ok can you tell me something about A² x , where x is an n*1 vector ?
Me :	( after 1 min of thinking ) consider A² (x) = A(Ax) = A(y) [ taking Ax = y ] when we multiply a Matrix with a vector we get other vector that is in a column space of that matrix. so here also we get Ax = y is a vector which is in a Column space of A but as we know that for a matrix A , Col space of A = Null space of A so Ax = y is a vector which is also in a Null space of A => y is in a null space of A so A(y) = zero vector, because when we multiply a matrix with a vector in a null space of that matrix. we get zero vector ( definition of null space ) so A² (x) = zero vector

I1 :	Yes correct, now can you tell me something about A² ?
Me :	i was thinking for 30-40 sec but didn't come with anything.

I1 :	take 1-2 examples and tell me what you observe.
Me :	i take 2*2 matrix A which i derived in the above question A = [1 -1] [1 -1] when i tried A² , i got zero matrix.

I1 :	can you give the reason for this ?
	[ with some hints given by I1 ]
Me :	i give a reason as follows : As we prove that : " A² (x) = zero vector , for any vector x " this can be true iff A² is a zero matrix. if A² is not a zero matrix then there exist vector x for which A² (x) = non zero vector.

I1 :	i have done with Linear Algebra.

	[ One of the faculty member said that he was going to ask a programming question ]

I2 :	" Write a function which takes two arguments 1) an array of size n and 2) integer value k. Your function should return the k^th smallest element from the array ".
Me :	I was thinking to use bubble sort but he said that do it without sorting an array. I know i can use HEAP and then do delete_min operation k times but i had implemented heap only one time during the btech and i know that at least i am not able to do it with C in this 10-12 min so i didn't told about that solution. I tried some diff strategies of O(n²) but stuck at some point and after 7-10 min he said OK you can try this after this Interview, it has a simple solution.

	After that Interview is over.

Conclusion :

Faculties who are taking interviews are very friendly and give some hints when we are stuck on a question.
I like some of the Linear Algebra questions very much they are really new questions.
I got selected for the MS HTRA program.

MS Interview Preparation

Sr.	Link	Description
1	Gilbert Strang video Lectures Advanced Matrix Theory by Prof. Vittal Rao (IISC) Video Lectures Problem practice for Linear Algebra	Linear Algebra Materials for Interview. [ 3rd link contains problems from the Top universities assignments very good site to practice problems topic wise on LA ]