Can someone do part 12 and 3 please Thanks Safari File Edit

Can someone do part 1,2, and 3 please! Thanks

Safari File Edit View History Bookmarks window Help BE 74% Tue 3:03 PM Q E mathcms2 asu.edu https:/fmath... Through He MAT 242 Sp MAT 242 Written Homework #3 EP4.2, 4.3 H4.2, 4.3 Due: March 15 Solve the following problems, showing any necessary work n Dash Then 1. [1 point] Let W be the subspace spanned by K Which of the following vectors are in W? (Note that the number of correct answers might be zero, one, two, three, or four.) r 71 15 r 20 r-201 15 -5 15 24 12 17 16 -6 L J 12 16 3 and iA -1 2. Let if -3 v2 v3 a. [1 point] The set tii, ii2, i3, i4 is linearly dependent. Find a nontrivial linear combination of these vectors that adds up to 0.

Solution

Ans(1):

We know that if {a,b} is basis of subspace W then for any u in W, we can write

u=ax+by where x and y are some real numbers.

same way if [7,-1,-24,-6] belongs to W then

[7,-1,-24,-6]=[-2,-1,2,-2]x+[1,3,1,-1]y  

[7,-1,-24,-6]=[-2x+y,-1x+3y,2x+y,-2x-1y]

comparing both sides we get

-2x+y=7...(i)

-1x+3y=-1...(ii)

2x+y=-24...(iii)

-2x-1y=-6...(iv)

solving (i) and (ii) gives x=-4.4 and y=-1.8

if those values of x and y satisfies equation (iii) and (iv) then [7,-1,-24,-6] belongs to W

check for (iii)

2x+y=-24

or

2(-4.4)+(-1.8)=-24

or -10.6=-24 which is FALSE

Hence [7,-1,-24,-6] doesn\'t belongs to W.

-----------------------------------------------

similarly we can check for remaining choices and find that only last choice satisfies all four equations for x=9 and y=-2

Hence only [-20,-15,16,-16] is in W.