A=
1 2 3 1
−2 −4 1 1
6 −1 −2 1
,c=
c 1
c 2
Let S = {(1, 2, 3), (−2, −4, 1), (6, −1, −2)}. (a) Determine if the set is affinely independent or dependent. If it is affinely depen- dent, find such a combination. (b) Describe conv(S) geometrically. Be as descriptive as possible. 1 (c) If another non-zero vector v is added to the set S, is it ever possible for this new set to be affinely independent? Explain
To determine the affine independence of the set ( S = {(1, 2, 3), (-2, -4, 1), (6, -1, -2)} ), we can follow these steps:
A set of points ( {p_1, p_2, \ldots, p_n} ) in ( \mathbb{R}^m ) is affinely independent if the vectors ( p_2 - p_1, p_3 - p_1, \ldots, p_n - p_1 ) are linearly independent. For three points in ( \mathbb{R}^3 ), they are affinely independent if the vectors formed by taking one point as a reference and the others as endpoints are not coplanar.
Choose a reference point: Let ( p_1 = (1, 2, 3) ), ( p_2 = (-2, -4, 1) ), and ( p_3 = (6, -1, -2) ).
Form the vectors: [ v_2 = p_2 - p_1 = (-2, -4, 1) - (1, 2, 3) = (-3, -6, -2) ] [ v_3 = p_3 - p_1 = (6, -1, -2) - (1, 2, 3) = (5, -3, -5) ]
Check for linear independence: We can check if the vectors ( v_2 ) and ( v_3 ) are linearly independent by forming a matrix and calculating its determinant: [ \begin{bmatrix} -3 & 5 \ -6 & -3 \ -2 & -5 \end{bmatrix} ]
To check for linear independence, we can compute the determinant of the matrix formed by these vectors. However, since we have only two vectors in ( \mathbb{R}^3 ), we can also check if one is a scalar multiple of the other.
We can set up the equation: [ k(-3, -6, -2) = (5, -3, -5) ] This gives us the system of equations: [ -3k = 5 \quad (1) ] [ -6k = -3 \quad (2) ] [ -2k = -5 \quad (3) ]
Solving equation (2): [ k = \frac{-3}{-6} = \frac{1}{2} ]
Substituting ( k = \frac{1}{2} ) into equation (1): [ -3 \left(\frac{1}{2}\right) = 5 \quad \text{(not true)} ]
Since the values of ( k ) do not match, the vectors are linearly independent. Therefore, the set ( S ) is affinely independent.
The convex hull ( \text{conv}(S) ) of the set ( S ) is the smallest convex set that contains all the points in ( S ). Since ( S ) consists of three affinely independent points in ( \mathbb{R}^3 ), the convex hull will be a triangle.
Geometrically, this triangle will be formed by the points ( (1, 2, 3) ), ( (-2, -4, 1) ), and ( (6, -1, -2) ). The triangle will lie in a plane in ( \mathbb{R}^3 ) defined by these three points. The area of this triangle can be calculated using the cross product of the vectors ( v_2 ) and ( v_3 ).
If we add another non-zero vector ( v ) to the set ( S ), the new set will be ( S' = S \cup {v} ).
For the new set ( S' ) to be affinely independent, the new vector ( v ) must not lie in the affine span of the original points in ( S ). Since ( S ) already consists of three points in ( \mathbb{R}^3 ), adding a fourth point can lead to affine dependence unless the new point does not lie in the plane formed by the original three points.
In general, if ( v ) is chosen such that it does not lie in the plane defined by the triangle formed by the points in ( S ), then it is possible for the new set ( S' ) to be affinely independent. However, if ( v ) lies in the plane, then the set will be affinely dependent.
In conclusion, it is possible for the new set to be affinely independent if the new vector ( v ) is not in the affine span of the original points.