Question

Let S= span { [2, 1], [4, 2] }. Determine if the vector [0,3] is in S. Determine if the vector [3,0] is in S. Determine if S= R². Justify your answer.

Answer 1

Yes, the vector [0,3] is in S, but [3,0] is not. Therefore, S is not equal to R^2.

Step-by-step explanation:

In order to determine if the vector [0,3] is in the span S, we need to check if it can be written as a linear combination of the vectors [2,1] and [4,2].

Let's set up the equation [0,3] = a[2,1] + b[4,2], where a and b are scalar coefficients.

By solving the system of equations, we find that a = 1 and b = 1/2. Therefore, [0,3] is in S.

To determine if the vector [3,0] is in S, we can use the same approach. By solving the equation [3,0] = a[2,1] + b[4,2], we find that there is no solution, indicating that [3,0] is not in S.

Since [0,3] is in S and [3,0] is not in S, S is not equal to R^2.