Question

1. What relationships are there between the rank/nullity and the eigenvalues/eigenvectors?

2. How do the eigenvalues and eigenvectors of each matrix change under any arbitrary change of basis?

3. What are some famous bases? What are they used for and why are they chosen for that particular application? (less than 500 words)

4. What is the point of linear algebra (less than 200 words)

Can you please help me out with this I am stuck and I am so confused how to start and how to end, can you help me with this. Take your time Thank You!

Answer 1

a matrite M is lequal to the eigenvalues of m and the Rhe filho. Sa The only selakonship between the sank and the eigensectos a matrise that Sean think of the followings- eigenvalue of nullity of the matrire. Also, the stant and the nullity add of to the number of column of M. Hence, the number of lineally independent eigen rectors of the zero rank of maddep to the number of Column of 2An Sha linear algebrug an egenvectos ar characteristic Vector & a linean Fransformation is a non zero vectos Hat changes at most by a scalar factor when that linean elgen value is the factors by the ecnich eigen vector is sales. eigenvectos gcorsesponding to a sheal non eigenvaheen ng pants in a direction in which it is strected the hans formation and He eigen value is He fachor By whichita Strected. If the eigenvalue is negative, the direction is reversed. 3 As Let A be the square matsion of order nan, then a number (Real or complers) A is said to be eigenvalue of makin A.- If there exista column males of order nen such Ax = x - Eigen vector of A is Geometrically an Zero hat L eigen Talue of A. Ax-dx=0 (And) x=0 working to find Eigen value and Eigen verloren

Solution of charadristic of equar is called Characteristic equation of matin A [1Adil of characteristic equation hy square matric. . eigen ار CU values Eigen vechora- dad, t ce taz- LA-d, I)x=0 vector of matin Corresponding to d=di. n Simplen, problems. For egin Of stress is applied to a "platih solid, the deformation can be dissected into principle directions" those direction in which deformation is In Wheer algebra, an ejen vector au characteristic ve oko de Linear transformation is a non provector that cho changes at most by a scalar factor They asle used for the Communication system: Hey Hep retical limit to how much information transmitted a communication medium like your telephone or the define the through This is air. The eigen vectors and through the

of the fundamental modes of the change themselves are eigenvalues of the communication channel (empressed a matris) and eigenvalues. The eigenvalues are Her, in essence, they eigenvectors. -caphragare naturally all the time, for ex. then ysetry Convenient case of lineas -- Unear algeboom Sequiffon's acige gratu when we have some equations that we care about we ofle ie. We want to know if there is at least one solution, and if So, we would like to panametize the set of all solutions in a unfortunate fact is that & beyond the equation the domain of becomes intractable, even using a computer solving systems of polynomial equations very quickly To explain, let us consider the case of equaris one o Variable a solution Solving a linear equation AxtBzo is something most elementary school students have the capacity for (even if they don't hearn the algebraic language. To solving a quadrabic Ax2 + Bxta 20 is part of the standard Comiculum; in fact there exist a closed formula for the two solution (in the set of In this terms of the.com complex numbers efficients equation in A, B, C.

desired Now, when there is just one equation infust Vasciable, Here's not the des degree of precision in an efficient way. Ifo rexi- Nowhero Raphson iteration be apponimat. He can be identi di manutely can