Question

Let u = [1, 2], v = [3, 4], and w = [5, 6]. Select true or false for each statement.

The vector w is in the image of u.
The vector v is in the kernel of u.

Answer 1

The vector w is not in the image of u. The vector v is not in the kernel of u.

Step-by-step explanation:

To determine if the vector w is in the image of u, we need to check if there exists a scalar k such that ku = w. If we multiply vector u by any scalar, the result will always have the same direction as u, but the magnitude may be different. Therefore, the vector w = [5, 6] cannot be obtained by multiplying u = [1, 2] by any scalar, so the statement 'The vector w is in the image of u' is false.

To determine if the vector v is in the kernel of u, we need to check if the dot product of u and v is equal to zero. The dot product of u = [1, 2] and v = [3, 4] is 1 * 3 + 2 * 4 = 11, which is not equal to zero. Therefore, the statement 'The vector v is in the kernel of u' is false.