Question

Let U={x(-R^3|x1+2x2-x3=0}
and V={x(-R^3|x1-2x2-2x3=0}
Find a basis of -
span (U or V)

Answer 1

Find a basis for U and a basis for V. Then combine the bases, being sure to eliminate any vectors in the union of the two bases that is a linear combination of the others.

Answer 2

Mr. James, I have formed a basis of U as follows, Can you check it please?

Basis of U = (1, -1, -1)^T, (1, 1, 3)^T, (1, 0, 1)^T

Answer 3

and Basis of V=(4, 1, 1)^T, (2, 2, 0)^T, (0, 1, -1)^T

Answer 4

No, U and V are not 3-dimensional subspaces of R^3. If they were, they would be both R^3.

Geometrically, they are both planes passing through the origin; hence they are both 2-d subspaces.

Answer 5

could suggest an example basis of U and V?

Answer 6

James could you give me an example of basis of U of the above problem?

Answer 7

For U we know that x1+2x2-x3=0. Hence one vector in U is (2,-1,0). Another is (1,0,1). They are clearly independent and if we accept that U is 2-D, those two vectors would be a basis.

Answer 8

ok, then, with 2-D , we should consider 2 subspace of R^3?

Answer 9

If a sub-space is two dimensional, then it has two basis vectors. Hence I'm saying for U, a basis is
{(2,-1,0), (1,0,1)}

Make sure you're convinced that both of those vectors are in U. I.e., that they satisfy the condition that defines U.

If you understand that, you will be able to find a basis for V by yourself.

Answer 10

Mr. JamesJ, since the two space defined as above intersects, the intersections has become 2 dimensional? I am still confused..