00:01
In this problem, we need to determine if the subset of r2 consisting of vectors of the form ab, where a and b are integers is a subspace, and we need to select true or false for each of the given statements.
00:13
So the first statement is this set contains the zero vector.
00:18
Now, note that the set contains all vectors of the form a, b, where a and b are integers.
00:27
Now, 0 is an integer so we can assign a the value 0 and b the value 0, so that 0 will belong to the set.
00:37
And 0 ,0, this is the 0 vector for r2.
00:40
So this set does indeed contain the 0 vector, and so this first statement will be true.
00:47
Now, the second statement is this set is a subspace.
00:51
Now, we will answer this question a little later.
00:54
Let us answer the other ones first.
00:56
Let us look at number three.
00:59
This set is closed under vector addition.
01:02
Now, let us consider two vectors from this set.
01:06
Let us consider the vectors to be a1, b1, and let us consider the vectors, the other vector to be a2, b2.
01:14
And if we add these two vectors, we will get a1 plus a2 and b1 plus b2.
01:21
Now, since these two vectors are in the set, so that means that a1, a2, b1, b2, all of them are integers.
01:29
And if a1 and a2 are integers, then a1 plus a2 is an integer.
01:33
And since b1 and b2 are integers, so b1 plus b2 is also an integer.
01:38
That means that this element will, this vector will belong to the set.
01:42
And so the set is indeed closed under vector addition.
01:45
So this statement will be true.
01:49
Next, let us consider the last statement...