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Vector transformation followed by coordinate point substitution and vice-versa, both given the same result. Choose the best answer.
- a)Possible, when the vector is constant
- b)Possible, when the vector is variable
- c)Possible in all cases
- d)Not possible
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Vector transformation followed by coordinate point substitution and vi...
The order of vector transformation and point substitution will not affect the result, only when the vector is a constant.
Most Upvoted Answer
Vector transformation followed by coordinate point substitution and vi...
Possible, when the vector is constant.
Explanation:
When we perform a vector transformation, we apply a linear operator to the vector, which can be represented by a matrix. The matrix represents the transformation that is applied to the vector.
When we substitute the coordinate point, we are essentially converting the vector from one coordinate system to another. This is done by multiplying the vector by a matrix that represents the change of basis between the two coordinate systems.
In order for the vector transformation followed by coordinate point substitution to give the same result, the following conditions must be met:
1. The vector must remain constant: If the vector changes its magnitude or direction during the transformation, the coordinate point substitution will not give the same result. Therefore, the vector must be constant for the two operations to be equivalent.
2. The matrix representing the vector transformation and the matrix representing the change of basis must be compatible: The dimensions of the two matrices must be such that their multiplication is valid. In other words, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
3. The matrices must satisfy the commutative property: This means that the order in which the operations are performed does not matter. If the vector transformation is performed first, followed by the coordinate point substitution, or vice versa, the result should be the same.
Given these conditions, it is possible for the vector transformation followed by coordinate point substitution to give the same result, but only when the vector is constant. If the vector is variable, the result will not be the same. Therefore, the correct answer is option 'A'.
Explanation:
When we perform a vector transformation, we apply a linear operator to the vector, which can be represented by a matrix. The matrix represents the transformation that is applied to the vector.
When we substitute the coordinate point, we are essentially converting the vector from one coordinate system to another. This is done by multiplying the vector by a matrix that represents the change of basis between the two coordinate systems.
In order for the vector transformation followed by coordinate point substitution to give the same result, the following conditions must be met:
1. The vector must remain constant: If the vector changes its magnitude or direction during the transformation, the coordinate point substitution will not give the same result. Therefore, the vector must be constant for the two operations to be equivalent.
2. The matrix representing the vector transformation and the matrix representing the change of basis must be compatible: The dimensions of the two matrices must be such that their multiplication is valid. In other words, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
3. The matrices must satisfy the commutative property: This means that the order in which the operations are performed does not matter. If the vector transformation is performed first, followed by the coordinate point substitution, or vice versa, the result should be the same.
Given these conditions, it is possible for the vector transformation followed by coordinate point substitution to give the same result, but only when the vector is constant. If the vector is variable, the result will not be the same. Therefore, the correct answer is option 'A'.
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