Advanced book on Mathematics Olympiad

2.3 Linear Algebra 79

243.LetAbe then×nmatrix whosei, jentry isi+jfor alli, j= 1 , 2 ,...,n. What

is the rank ofA?

244.For integersn≥2 and 0≤k≤n−2, compute the determinant

∣

∣∣

∣∣

∣∣

∣∣

∣

∣

1 k 2 k 3 k ··· nk

2 k 3 k 4 k ··· (n+ 1 )k

3 k 4 k 5 k ··· (n+ 2 )k

nk(n+ 1 )k(n+ 2 )k···( 2 n− 1 )k

∣

∣∣

∣∣

∣∣

∣∣

∣

∣

245.LetVbe a vector space and letf, f 1 ,f 2 ,...,fnbe linear maps fromV toR.

Suppose thatf(x)=0 wheneverf 1 (x)=f 2 (x)= ··· =fn(x)=0. Prove thatf

is a linear combination off 1 ,f 2 ,...,fn.

246.Given a setSof 2n−1 different irrational numbers,n≥1, prove that there existn

distinct elementsx 1 ,x 2 ,...,xn∈Ssuch that for all nonnegative rational numbers

a 1 ,a 2 ,...,anwitha 1 +a 2 +···+an>0, the numbera 1 x 1 +a 2 x 2 +···+anxn

is irrational.

247.There are given 2n+1 real numbers,n≥1, with the property that whenever one

of them is removed, the remaining 2ncan be split into two sets ofnelements that

have the same sum of elements. Prove that all the numbers are equal.

2.3.6 Linear Transformations, Eigenvalues, Eigenvectors............

A linear transformation between vector spaces is a mapT : V → W that satisfies
T(α 1 v 1 +α 2 v 2 )=α 1 T(v 1 )+α 2 T(v 2 )for any scalarsα 1 ,α 2 and vectorsv 1 ,v 2. A matrix
Adefines a linear transformation byv→Av, and any linear transformation between
finite-dimensional vector spaces with specified bases is of this form. An eigenvalue of a
matrixAis a zero of the characteristic polynomialPA(λ)=det(λIn−A). Alternatively,
it is a scalarλfor which the equationAv=λvhas a nontrivial solutionv. In this case
vis called an eigenvector of the eigenvalueλ.Ifλ 1 ,λ 2 ,...,λmare distinct eigenvalues
andv 1 ,v 2 ,...,vmare corresponding eigenvectors, thenv 1 ,v 2 ,...,vmare linearly inde-
pendent. Moreover, if the matrixAis Hermitian, meaning thatAis equal to its transpose
conjugate, thenv 1 ,v 2 ,...,vmmay be chosen to be pairwise orthogonal.
The set of eigenvalues of a matrix is called its spectrum. The reason for this name
is that in quantum mechanics, observable quantities are modelled by matrices. Physical
spectra, such as the emission spectrum of the hydrogen atom, become spectra of matrices.
Among all results in spectral theory we stopped at the spectral mapping theorem, mainly
because we want to bring to your attention the method used in the proof.