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first the definition of the rank of a matrix is "maximal number of linearly independent column vectors in the matrix"

then the question could be rephrased to " how many ind...

First, let's think of how many different possibilities we have to seat 6 people at a 'normal' table, i.e. in a straight line. Let's call the people A, B, C, D, E, F. We now have 6 seats:

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(bx+d)(bx-d)=b^2x^2-d^2(ax-c)(bx+d)(bx-d)=(ax-c)(b^2x^2-d^2)=ab^2x^3-ad^2x-b^2cx^2+cd^2ab^2=3b^2c=2ad^2=147-cd^2=98From equations:a=3/b^2c=2/b^2d^2=49b^2Since a, b, c, d are positive integers, b must be 1...

The procedure of mathematical induction is as following:

Firstly prove the base case is true i.e. when n=1, the statement is true.

Then assume for some integer n=k the state...

Here we are examining the imaginary number ‘i’, defined as the square root (sqrt) of ‘-1’.We begin by using Euler’s identity:e^(iπ)+1=0e^(iπ)=-1Since sqrt(x) is the same as x^(1/2):(e^(iπ))^(...