00:01
So in these questions, we're given two systems to essentially solve.
00:05
And we want to show that the set of solutions is a subspace of r3 in the first case and of r4 in the second case and give a basis in the dimension.
00:14
So let's just work through the systems, right? so for example, the first system, let's solve the first equation for x2, say.
00:22
So x2 is 2x3 minus x1, which we replace at the bottom by writing.
00:28
So 2x1 plus x2, which is 2x3 minus x1 equals, i'm going to put this minus 5x3 on the other side.
00:40
5x3.
00:42
So the first equation we leave the same.
00:44
And here we simplify to get 2x1 minus x1.
00:48
So x1 equals 5x3 minus 2x3.
00:52
So 3x3.
00:55
And actually we can go back to the first equation to replace.
00:59
Then x2 will be 2 x3 minus x1 x1 is 3 x3 so it is minus x3 right so the set of solutions of this system is a set of points we're writing everything as a function of x3 so the set of points x1 is 3 x3 x2 is minus x3 and x3 is free right where x3 is a real number and i'm going to make a small trick by noticing that if i factor out the x3, this is the set of multiples of the vector 3 minus 1 1 with x3 in r.
01:44
So what we're saying is that s is a space generated by the multiples of this vector, all possible multiples.
01:51
So the conclusion is that s is a subspace generated by the vector 3 minus 1 .1.
01:58
So it has dimension is equal to 1 and the basis is the vector 3 minus 1.
02:11
So this is a solution to the first question.
02:14
We're going to do exactly the same for the last question.
02:16
All right.
02:17
So here now we're going to just have a bit more calculations.
02:20
So the first one we're going to write x1 will be minus x2 minus x3 plus 5x4.
02:28
In the second equation we're going to write.
02:31
X2 as being minus x3 plus 3x4.
02:37
And now we substitute, right? so the first one will be minus 2x1.
02:40
So we get 2x2 plus 2x3 minus 10x4.
02:48
Then we have minus x2.
02:51
So plus x3 minus 3x4.
02:55
And then we have minus x3 and we have plus 7x4 equal to 0.
03:02
So let's be careful with the calculations to make sure don't make any mistakes...