Final answer:
To show that v1 does not belong to the span of v2 and v3, we can perform Gaussian elimination, row reduction, use the inverse matrix method, or apply Cramer's rule. Each method will determine if there is a solution to the system of equations or not.
Step-by-step explanation:
To show that v1 does not belong to the span of v2 and v3, we need to show that there are no scalars a and b such that v1 = av2 + bv3. To do this, we can express this as a system of equations:
a*v2 + b*v3 = v1
Performing Gaussian elimination on this system will allow us to determine if there is a solution or not. If the system becomes inconsistent, then v1 does not belong to the span of v2 and v3. If it becomes consistent, there is a solution and v1 does belong to the span of v2 and v3.
Alternatively, we can perform row reduction on the augmented matrix obtained from the system of equations. If the result is a row of zeros, the system is inconsistent and v1 does not belong to the span of v2 and v3. If there is a row without zeros, the system is consistent and v1 does belong to the span of v2 and v3.
We can also use the inverse matrix method to determine if v1 belongs to the span of v2 and v3. If the inverse of the matrix formed by v2 and v3 exists, then v1 belongs to the span. If the inverse does not exist, v1 does not belong to the span.
Cramer's rule can also be used to determine if v1 belongs to the span of v2 and v3. If the determinant of the matrix formed by v2 and v3 is non-zero, then there is a unique solution and v1 belongs to the span. If the determinant is zero, there either is no solution or infinitely many solutions, and v1 does not belong to the span.