00:01
Okay, so we're given three vectors u, v and w, which we're told are linearly independent, and we want to work out if the vectors v minus u, w minus v and u minus v are linearly independent.
00:13
So remember to show three vectors are linearly independent, we should show that the only way you can write them as a linear combination equal to zero is if all the coefficients are zero.
00:24
So what i mean by this is we want to take a linear combination of these three vectors, let's say a times v minus u plus b times w minus v plus c times u minus w we want this to be equal to the zero vector so if they're linearly independent the only way we can do this is if a b and c are all zero if they are linearly dependent then there exists non -zero a b and c such that this can be done so let's see if there exists non -zero a, b and c, or if the only way we can do this is for a, b, and c to be zero.
01:07
So we're going to group together the terms with the vs, they group together the terms with u's, and group together the terms with ws, and we're going to try and use the fact that the set uvw are linearly independent.
01:21
So if we group together the v terms, we have a, so that's this term, this term here a minus b times the vector v so we've done this and this now let's group together the u terms so this is minus a that's this term plus c times u these two terms and finally group together the w terms we end up with b minus c so that's these two terms times w equals the zero vector now since we know that v you and w are linearly independent this implies that a minus b is equal to zero c minus a is equal to zero and b minus c is equal to zero so a minus b equals zero c minus a equals zero and b minus c equals zero so all i've done here is use the fact that these three vectors are linearly independent.
02:33
So this means that a is equal to b, also a is equal to c, and also b is equal to c.
02:40
So a, b and c are all the same...