Consider (1/2)(u1-u2)=(0,1,0) and (1/2)(u1+u2)=(1,0,1)physicsss said:I'm stuck on the following problem:
Describe the span of the vectors u1 and u2 in R^3, where
u1 = (1, 1, 1), u2 = (1, −1, 1)
I know that the span is a(u1)+b(u2), which becomes (a+b,a-b,a+b), but I don't know where to go from here.
TIA.
The span of u1 and u2 in R^3 refers to the set of all possible linear combinations of the vectors u1 and u2 in three-dimensional space.
To find the span of u1 and u2 in R^3, you can create a matrix using the coefficients of the linear combination and use Gaussian elimination to determine the pivot columns. The vectors corresponding to the pivot columns will form a basis for the span of u1 and u2.
Finding the span of u1 and u2 in R^3 allows us to understand the range or column space of a linear transformation defined by these vectors. It also helps us determine if the vectors are linearly independent or if they span the entire space.
Yes, it is possible for the span of u1 and u2 in R^3 to be a plane or a line. This will depend on the linear combination of the two vectors. For example, if one vector is a scalar multiple of the other, the span will be a line. If the two vectors are linearly independent, the span will be a plane.
No, the span of u1 and u2 in R^3 cannot be the entire space. This is because the two vectors are only two-dimensional and cannot span a three-dimensional space. The span will always be a subspace of R^3.