Question

Find a basis containing
the vectors
(1,0,0) and (1,1,0) for the vector

Space R cube​

Answer 1

Answer:

The collection { i+j, j+k} is not a basis for R 3. Although it is linearly independent, it does not span all of R 3. For example, there exists no linear combination of i + j and j + k that equals i + j + k.

Example 4: The collection { i + j, i − j} is a basis for R 2. First, it is linearly independent, since neither i + j nor i − j is a multiple of the other. Second, it spans all of R 2 because every vector in R 2 can be expressed as a linear combination of i + j and i − j. Specifically, if a i + b j is any vector in R 2, then if k 1 = ½( a + b) and k 2 = ½( a − b).