Final answer:
The question asks to determine if certain vectors are in the image or kernel of a matrix. It involves solving linear equations to see if a solution exists for Ax = b, where A is the matrix and b is the vector in question.
Step-by-step explanation:
The subject of this question is Matrix Algebra, a topic within Mathematics, and it is likely targeted at the High School level. When determining if a vector is in the image of a matrix, we are essentially asking if there exists a vector x such that the matrix-vector product Ax equals the vector in question. This involves solving a set of linear equations. The kernel (or null space) refers to all vectors that are mapped to the zero vector by the transformation represented by the matrix. To check if a vector is in the image, we set up an equation Ax = b, with b being the vector we're investigating, and attempt to find a solution.
As the specific matrix A and vectors are not provided, we cannot demonstrate the exact process, but it would involve either using techniques such as Gaussian elimination to solve the resulting system of equations, or checking if the determinant of the matrix is nonzero (which would imply that the matrix is invertible, and thus every vector in the corresponding space is in the image).