Linear Algebra and Applications
alsteve69
test_4.pdf
Math 10 Test 4 Show all work/reasoning to receive full credit. 1. Given ),4,1,3(),3,1,2( 21 −== vv and )4,6,2(3 =v .
(a) What is the dimension of span },,{ 321 vvv ?
(b) Give a geometric description of span },,{ 321 vvv . 2. Find a vector that can be added to the set }1,1{ 2 −+ tt to form a basis for 2P .
3. Consider the matrix
− −
− =
031
0124
062
B . Find the null space of B and find nullity(B).
4. Let A be an nm × matrix. If B is an invertible mm × matrix, show that BA and A have the same nullspace
and hence the same rank. 5. Find the standard matrix for the composition in 2R : A reflection over the x-axis, followed by an orthogonal
projection onto the x-axis, followed by a rotation of °150 . 6. Consider the Transformation 33: RRT → , given by
),,2(),,( 32321321321 xxxxxxxxxxxT +−−+−=
Is T a matrix transformation? Justify your answer. If it is so, find the standard matrix. 7. Determine which of the following matrices have the same row space. Justify your answer.
− −−
= 543
121 A
− −
= 132
211 B
− − −
= 153
1012
311
C
8. Let
− −−−
−− −
=
120930
31112
31310
01201
A
Is )1,2,0,3,2( −− in the row space of A? Justify your answer.
9. Explain why the row vectors of a 34 × matrix form a linearly dependent set. 10. Let
),(),,( zxzyxzyxT ++−= and ),,(),( yyxxyxS −= Find TS o and ST o whenever defined.
11. Let T be the transformation from 2R into 2R such that T(u) = projv u where v = (1, 1). Find ),( yxT .
