LINEAR ALGEBRA EXAM 3 version 1 name: 1. Use eigenmagic to solve the following system of differential equations: dx dt = −4x− y − 2 z dy dt = 5 y dz dt = 6x+ 3 z o o A y(t) = c1e(8 t) + c2et + 72 c3 B...

Please see the Attachment.


LINEAR ALGEBRA EXAM 3 version 1 name: 1. Use eigenmagic to solve the following system of differential equations: dx dt = −4x− y − 2 z dy dt = 5 y dz dt = 6x+ 3 z o o A y(t) = c1e(8 t) + c2et + 72 c3 B x(t) = c3 C x(t) = c1e(5 t) + c3e(−t) + c2 D y(t) = −15 c1e(5 t) E z(t) = c1e(8 t) − 12 c3 F z(t) = 3 c1e(5 t) − 32 c3e (−t) − 2 c2 G None of These 2. Given A =  5 3 40 8 4 7 0 −4 , find a matrix P such that P−1AP is a diagonal matrix o o A P =  1 1 1−1 −25 −1 −8 −85 −1  B P =  1 1 0− 115 59 1 1 5 −1 0  C P =  1 1 11 −2827 1 1 2 7 9 − 7 2  D P =  0 1 11 −2 −207 0 1 2  E None Of These 3. Suppose linear transformation T : R2 → R2 with T [ −5 1 ] = [ 1 0 ] and T [ 2 −1 ] = [ −1 3 ] Select the true statement/s o o A T [ 1 0 ] = [ −13 2 ] T [ 0 −1 ] = [ 14 3 −25 ] B T [ 1 0 ] = [ 0 −1 ] T [ 0 −1 ] = [ −1 5 ] C None of These 4. Suppose V = {[ a a ] : a ∈ R } with the standard addition and the standard scalar multiplication by real numbers. Which Ax- ioms are satisfied? o o A IDENTITY LAW for SCALARS holds B CLOSURE LAW for SCALARS holds C ASSOCIATIVE LAW for SCALARS holds D V is a COMMUTATIVE GROUP E DISTRIBUTIVE LAW for scalars holds (scalars distribute) F ALL 10 axioms hold, V is a vectors space over R G DISTRIBUTIVE LAW for vectors holds (vectors distribute) H None of These 5. Consider the following system of equations. Then determine the true statement/s  1 1 1 01 1 1 0 2 −5 1 −5  ·  x y z w  =  44 −3  o o ©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 1 of 3 http://www.daabz.com/editQ.html?nID=2jw http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=1&key=3 http://www.daabz.com/editQ.html?nID=2jI http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=2&key=3 http://www.daabz.com/editQ.html?nID=2j6 http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=3&key=3 http://www.daabz.com/editQ.html?nID=2hQ http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=4&key=3 http://www.daabz.com/editQ.html?nID=2i http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=5&key=3 LINEAR ALGEBRA EXAM 3 version 1 (page 2/ 3) A The Homogenous solutions are given by x⃗h = α  1 0 0 1 + β  0 1 1 1  B ALL solutions are of the form x⃗ = x⃗p + x⃗h C The Homogenous solutions are given by x⃗h = α  1 1 −2 −1 + β  0 5 −5 −6  D A particular solutions is given by x⃗p =  3 0 1 2  E ALL solutions are of the form x⃗ =  3 0 1 2 + α  1 1 −2 −1 + β  0 5 −5 −6  F None of These 6. Determine the true statement/s about matrixM where M =  1 −6 3 21 0 −1 −1 −1 0 1 1  o o A A basis for the row space ofM is given by the non-zero rows in  1 0 0 −210 1 0 2 0 0 4 −17  B A basis for the column space ofM is given by the non-zero columns in  1 0 0 00 1 0 0 0 −1 0 0  C A basis for the column space ofM is given by the non-zero columns in  1 0 0 00 1 0 0 0 0 1 0  D rank ofM is 4 E rank ofM is 2 F rank ofM is 3 G A basis for the row space ofM is given by the non-zero rows in  1 0 −1 −10 6 −4 −3 0 0 0 0  H rank ofM is 1 I None of These 7. SupposeB = {( −2 −2 ) , ( 4 4 )} and V be the space spanned by linear combinations of B using real number scalars. In other words V = { α ( −2 −2 ) + β ( 4 4 ) : α, β ∈ R } with the standard addition and the standard scalar multipli- cation. Select true statement/s o o A V is a vector space over R B ( −18 −18 ) ∈ V C ( 0 −1 ) ∈ V D ( 1 −3 ) ∈ V E ( 1 3 ) ∈ V F None of These 8. Suppose V is defined to be the space spanned by the column of the matrix A =  0 1 21 −3 −1 3 1 −1 . Find the orthonormal basis for the space, and construct a matrix out of such basis as columns. o o A  0 111 √ 11 5 √ 2 55 1 10 √ 10 − 311 √ 11 32 √ 2 55 3 10 √ 10 111 √ 11 −12 √ 2 55  B  0 1 18111 −3 2755 3 1 − 955  ©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 2 of 3 http://www.daabz.com/editQ.html?nID=2im http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=6&key=3 http://www.daabz.com/editQ.html?nID=2hY http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=7&key=3 http://www.daabz.com/editQ.html?nID=2iK http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=8&key=3 LINEAR ALGEBRA EXAM 3 version 1 (page 3/ 3) C  0 1 17111 −3 4255 3 1 − 1455  D  85 167 √ 167 55 1 11 √ 11 5 √ 2 55 42 167 √ 167 55 − 3 11 √ 11 32 √ 2 55 − 14167 √ 167 55 1 11 √ 11 −12 √ 2 55  E none of these 9. v⃗ · v⃗ = ||v⃗||2 o o A false B true ©2019 daabz.com LINEAR ALGEBRA EXAM 3 version 1 page 3 of 3 http://www.daabz.com/editQ.html?nID=GH http://www.daabz.com/editQn.html?coursenid=1N&v=1&TaskType=exam&QuesNo=9&key=3