Your answers are correct for (i) and (ii).
A set of three vectors spans if and only if they are linearly independent over . Do you know how to find out if a set of vectors is linearly independent?
Determine (giving reasons for your answer) whether or
not the following subsets ofR3 are subspaces of R3:
(i) {(x, y, z) | xy = 0}
(ii) {(x, y, z) | x = 3z}
for (i) I got no, the set is not a subset of R3
for (ii) Yes, the set is a subset of R3
Could someone please tell me if I'm right or not...?
What is thespan of a set of vectors in Rn? Does the set of vectors
{(1, 2, 4), (2, 1, 3), (4,−1, 1)} span R3? Give reasons for your answer.
Would someone be able to step through this problem for me please...?
yes i think so... I just want to see if the set spans R3 not if it's linearly independent. I thought I would only need to check if it's linearly independent if I was lookin for a basis of R3
i Know that you can write any element in R3 as a combonation of S (where S is equal to the set)
e.g. (x,y,z) = a1(1,2,4) + a2(2,1,3) + a3(4,-1,1)
=> x = a1 + 2a2 + 4a3
y = 2a1 + a2 - a3
z = 4a1 + 3a2 + a3
It's at this stage I get stuck... I think I am ment to write a1, a2, a3 in terms of x, y, z...?
I'm not sure if im able to do this or if this is even the right method
Any help would be much appreciated
x = a1 + 2a2 + 4a3
y = 2a1 + a2 - a3
z = 4a1 + 3a2 + a3
x = a1 + 2a2 + 4a3
y-2x = 0 - 3a2 - 9a3 R2-> R2 - 2R1
z-2y = 0 + a2 + 3a3 R3 -> R3 - 2R2
x = a1 + 2a2 + 4a3
y-2x = 0 - 3a2 - 9a3
(z-2y) + 1/3(y-2x) = 0 R3-> R3 + 1/3R2
Any ideas were to go from here...? I'm sorry for all tge questions, it's just it's really frustrating getting stuck mid-problem
It is simpler to throw out the "a"s and just do the Gaussian elimination on the matrix of coefficients:
Subtract twice the first row from the second row and subtract 4 times the first row from the third row:
Subtract 3 times the second row from the third
.
There you have it: Since the last row is all 0s those vectors are not independent. Since no two are multiples of each other, take any two of the three matrices as basis.