Question

Let a1 = [1] , a2 = [-6] , and b = [5] For what value(s) of h is b in the plane spanned by a1 and a2?

[2] [-6] [-2]
[-1] [2] [h]

Answer 1

To find the value of h that puts vector b in the span of vectors a1 and a2, solve the system of linear equations x - 6y = 5 and 2x - y = h. The vector b is in the plane for any h that is 10 more than 11 times some real number y.

Step-by-step explanation:

To determine for what value(s) of h the vector b is in the plane spanned by a1 and a2, we need to find if there are scalars x and y such that x * a1 + y * a2 = b. Since a1 = [1, 2], a2 = [-6, -1], and b = [5, h], we have the following system of linear equations:

• x - 6y = 5
• 2x - y = h

By solving this system, we can find the values of x, y, and h that satisfy both equations, thus placing b within the plane of a1 and a2. If no solution exists, then vector b is not in the span of a1 and a2.

Example solution:

1. First equation: x = 5 + 6y
2. Second equation: 2(5 + 6y) - y = h
3. Simplifying, we get 10 + 11y = h
4. Thus, h can be any number that is 10 more than 11 times a number y. This means for any real number y, there will be a corresponding h that puts b in the plane of a1 and a2.