- #1
cookiesyum
- 78
- 0
Let V = R^3. Let W be the space generated by w = (1, 0, 0), and let U be the subspace generated by u_1 = (1, 1, 0) and u_2 = (0, 1, 1). Show that V is the direct sum of W and U.
Last edited:
Dick said:Try and follow the forum guidelines, ok? Show an attempt or state what's confusing you. Don't just post the question. You want to show each vector in V=R^3 can be expressed as the sum of a vector in U and a vector in W in a unique way. Try it.
In linear algebra, the direct sum of two vector spaces V and W is a third vector space that contains all possible combinations of vectors from V and W. In other words, it is a way of combining two vector spaces without overlapping any of their elements.
To show that U and W are direct sum of V, you need to prove that their sum is equal to V and their intersection is equal to the zero vector. This means that every vector in V can be written as a unique combination of vectors from U and W, and U and W do not share any common vectors.
There are two conditions that must be met for U and W to be direct sum of V. Firstly, the sum of U and W must be equal to V. Secondly, the intersection of U and W must be equal to the zero vector. If both of these conditions are satisfied, then U and W are direct sum of V.
Yes, a vector space can have more than two direct summands. In general, a vector space can have any number of direct summands as long as the sum of all the direct summands is equal to the original vector space.
To prove that two vector spaces are direct sum of each other, you need to show that their sum is equal to the original vector space and their intersection is equal to the zero vector. This can be done by finding a basis for each vector space and showing that they are linearly independent and span the original vector space.