Question

2 -25 4)[10+10+10pts.) a) Find the eigenvalues and the corresponding eigenvectors of the matrix A = b) Find the projection of the vector 7 = (1, 3, 5) on the vector i = (2,0,1). c) Determine whether the given set of vectors are linearly independent or linearly dependent in R" i) {(2,-1,5), (1,3,-4), (-3,-9,12) } ii) {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) }

Answer 1

4. Here the matrix a 1.3] Eigen value of A is obtained by det (A-x) =0 2 sy 2 5-) > 27 (1-x) (5-1) + 4-0 ☆ - 67+5+4=0 (x-3)=0 27 = 3,3 ère x=3 is the eigers value of a with multiplicity 2. Now, for a=3, (A-1)=0-> - 3 2 - 2 5-3 considering augmented motrin, ( - -2 2. -1 - 2 2 o O R RR₂ . So 1 in the reduced echelon foom, and colonn es alithout Reading entries 2nd column in now - pinotal column, de 'es free variable. So, system & - th 20 sy ha az so can be written an for = 2, v. [1] as an eigen vector correspondings X=-3

(b) Inner product two vector in inner product space IR³ geven (ie dost product). <ū, '> - ū. Here = (1,3,5 and ů = (2,0,0) NOW, - Bū - (1,3,5), (2,0,1) - 2 tot 5 = 7. and <u, uz ū. ū. (2,0, 1). (2,0,1) 4 +0+1 -5 The projection of ñ Т» is geven by <2, n> Tū> ū - 7 um 2,0,1) pooja B (0) Proton (ن Here, we can The given set {(2,-4, 5), (1,3,-4), (-3,-9, 12) write the vector (-3,-9, 12.) as lénean combination of other two rector as a Slow (-3,- 9, 12) = 0: (2,-1, 5)-3(1,3,-4) The set of nec vector is rin linearly dependent. -

Now, we w write ☺ the set {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)] Zero vector (0,0,0,0) as linean combination of the given set anita scolar a, b,c,d. (0,0,0,0) = a (1,0,0,0) + b(0,1,0,0)+ c(0,0,1,0) +4 (0,0,0,1) (0,0,0,0) = (a, b, c,;-) only solution is ire azo, bro, cao, do Therefore zero vector (0,0,0,0) can be written as Zero, Linear combenation of the nectors in the set only of all scolars are inearly indepenolent. Hence the set ês lin