- #1
charlies1902
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Suppose that S = {v1, v2, v3} is a basis for a
vector space V.
a. Determine whether the set T = {v1, v1 +
v2, v1 + v2 + v3} is a basis for V.
b. Determine whether the set
W = {−v2 + v3, 3v1 + 2v2 + v3, v1 −
v2 + 2v3} is a basis for V.
I must check if they're linearly independent.
For a:
c1v1+c2v1+c2v2+c3v1+c3v2+c3v3=0 c's are constants
v1(c1+c2)+v2(c2+c3)+v3(c3)
Forming the matrix gives
1 1 0
0 1 1
0 0 1
rref of this matrix is the identity matrix, thus it's linearly independent.
For b:
the same thing was done except the rref of the matrix was not the identity matrix, thus it's not a basis.
My question is is there an easier way to do this problem? It seems i made it harde/longer.
vector space V.
a. Determine whether the set T = {v1, v1 +
v2, v1 + v2 + v3} is a basis for V.
b. Determine whether the set
W = {−v2 + v3, 3v1 + 2v2 + v3, v1 −
v2 + 2v3} is a basis for V.
I must check if they're linearly independent.
For a:
c1v1+c2v1+c2v2+c3v1+c3v2+c3v3=0 c's are constants
v1(c1+c2)+v2(c2+c3)+v3(c3)
Forming the matrix gives
1 1 0
0 1 1
0 0 1
rref of this matrix is the identity matrix, thus it's linearly independent.
For b:
the same thing was done except the rref of the matrix was not the identity matrix, thus it's not a basis.
My question is is there an easier way to do this problem? It seems i made it harde/longer.