Question

heoWoing pradlen are so be homed on on Werbsenday: p 17. Show all wark ant insdicate any lememary roe opsrations ased in sofation of syseses, caleuloion of ieweres.ete. m written in PENCIL i our own hand Skow esough work to jumty your asuners Write th reporate shevts of psper which you alach do nov wowk she probiens on th age. No peinted horwork or homewowk copved fom a solulons mamal or onine source wii be graded wnt so wre your howork to heip you stoty, make a copy befare you hand ir ie All problems are to be prabens on . a) Find a ,utset of the set ofwotors (v.-(-1, 0.2),Y; basis for the suhepace of R* spenned by these vectoes (1.1,1)尚-(0,1,1), e.-(2.1,0)} that forms (b) Express each vector not in the beeis as a liner combination of the basis veciors 2. Find a 3x3 matrix whose null space is a) a point. b) a lise. c) a plane in each cuse, show why your answer works. 2 13 -1 5 2 13 -1 3. Find the rank and nullity of the malrix A 3 by reduxing it to sww schelon form. 4. Let A be a B x 5 matrix such that Ax-0 has only the trivial solution. Find the rank and mulcy of A. s. Find the characteristic equation, the eigenvalues, and bases for the eigenspoces of the matrix 2 3 6. Find the characteristic equation, the sigenvalues, and bases for the eigenspaces of the matris A-1 1 1 7. Prove ordsprove: lf A is an π × n matrix with real entries and n is odd, A has at least one real eigenvalue

Question 4

Answer 1

A is a 8 times 5 matrix Since A x = 0 has only trivial solution, Kernel of A = {0} is dimension of kernel (A) = 0 i.e. Nullity of A = 0 Now, by rank- nullity theorem, rank (A) + nullity (A) = 5 implies rank (A) + 0 = 5 implies rank (A) = 5 therefore rank of A = 5 and nullity of A = 0