Let a subset w be the set of all vectors in r4 such that...

Let a subset W be the set of all vectors in R^4 such that x1 + x4 = x2 + x3. Apply the theorem for the condition for a subspace to determine whether or not W is a subspace of R^4. According to the theorem of conditions for a subspace, the nonempty subset W of vector space V is a subspace of V if and only if it satisfies one of the following two conditions: 1.) If u and v are vectors in W, then u + v is also in W. 2.) If u is in W and c is a scalar, then the vector cu is also in W.

Transcript

00:01 Okay, so we have this set of vectors in r4, w, which is the set of vectors of the form x1, x2, x3, x4, which satisfy this equality.

00:11 So x1 plus x4 is equal to x2 plus x3.

00:15 So the first and fourth components added up are equal to the sum of the second and third components.

00:21 We want to check whether or not this is a subspace of r4.

00:24 So we need to show, if it is a subspace, we need to show that the sum of two vectors is in the space, is in the set.

00:31 And a scale and multiplication is in the set.

00:34 So let's let say u and v being w.

00:39 So if we take u1, u2, u3, u4, and v to be v1, v2, v3, v4, well, we know that u and v are both in w, so their components satisfy this condition.

00:58 So we know that u1 plus u4 is equal to u2 plus u3 and same for v.

01:05 V1 plus v4 is equal to v2 plus v3 we want to show that so we want to show that the u plus v is in w so let's write this vector u plus v we just add the vectors component -wise so u1 plus v1 u2 plus v2 u3 u3 plus v3 u4 plus v4 we want to show that the sum of the first and final components are equal to the sum of these two components.

01:38 Well, let's do this.

01:39 So u1 plus v1 plus u4 plus v4.

01:45 So this should be equal to this plus this so we can collect the u terms by rearranging this is u1 plus u4 plus v1 plus v4.

01:57 And now using the fact that u and v are both in w, this first bracket is, is u2 plus u3 the second is v2 plus v3 and then rearranging again this is u2 plus v2 plus v3 so indeed this component plus this is equal to this plus this so we've shown that u plus v is in the set w next we need to show that scale and multiplication is closed the set is closed under scalar multiplication...

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