Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span.S = {(−1, 2), (2, −1), (1, 1)}S spans R2.S does not span R2. S spans a line in R2. S does not span R2. S spans a point in R2.
Question
Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span.S = {(−1, 2), (2, −1), (1, 1)}S spans R2.S does not span R2. S spans a line in R2. S does not span R2. S spans a point in R2.
Solution
To determine whether the set S spans R2, we need to check if the vectors in S can be combined to form any vector in R2.
Step 1: Write down the vectors in S. S = {(−1, 2), (2, −1), (1, 1)}
Step 2: Set up a system of linear equations. We want to find scalars a, b, and c such that for any vector (x, y) in R2, a*(-1, 2) + b*(2, -1) + c*(1, 1) = (x, y)
This gives us the system of equations: -1a + 2b + c = x 2a - b + c = y
Step 3: Solve the system of equations. This system of equations has a solution for any (x, y) in R2 if and only if the determinant of the coefficient matrix is not zero.
The coefficient matrix is [[-1, 2, 1], [2, -1, 1]]
The determinant of this 2x2 matrix is (-1)(-1) - 22 = 1 - 4 = -3, which is not zero.
Therefore, the set S spans R2.
If the determinant had been zero, then S would not span R2. In that case, we would need to find the rank of the matrix to determine the geometric description of the subspace that S spans. The rank of a matrix is the maximum number of linearly independent rows or columns. If the rank is 1, then S spans a line in R2. If the rank is 0, then S spans a point in R2.
Similar Questions
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