00:01
In this question, we are given that the vectors u, v, and w are linearly independent.
00:08
And we are asked to show that u plus v, u minus v, and u minus 2 v plus w are also linearly independent.
00:17
First of all, let's recall what it means for vectors to be linearly independent.
00:22
This simply means that the linear combination c1u plus c2v plus c3w equal 0 as only trivial solutions, meaning that c1 equals c2 equals c3 is equal to 0.
00:44
Now let's create a linear combination for u plus v.
00:52
Let's call it not c1 but d1 times u plus v plus d2 times u minus v plus d3, plus d2 times u minus v, plus d3, times u minus 2v plus w equal 0.
01:18
To show that these vectors are linearly independent, we want to show that this linear combination is equal to 0, only when the 1, d2, and d3 are equal to 0.
01:30
Now let's multiply u plus v by d1.
01:37
We are going to get d1u plus d1 v plus d2u minus d2 v plus d3 v plus d3 2d3v plus d3w equals 0.
02:00
Now let's combine terms with u, terms with v and terms with w together.
02:08
We're going to get.
02:10
So terms with u are d1 u and d2 u and d3 u.
02:18
Terms with v are d1v and negative 2 d3v right in terms with with w only d3 w so for u 1 we're going to get d1 plus d2 plus d3 times u plus d1 plus d1 minus d2 minus 2 d3 times v plus d3 times v plus d3 times w is equal to 0 now we know that each of we know that the vectors u, v, and w are linearly independent, meaning this linear combination has only trivial solutions.
03:07
And this means that d1 plus d2 plus d3 has to be equal to zero, as well as d1 minus d2 minus 2 d3, and d3 also has to be equal to zero.
03:26
All right...