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Is the transformation T(x, y, z) = (x, y, 0) linear?
- a)No, it is not linear because all z components map to 0
- b)No, it is not linear because it does not satisfy the scalar multiplication property,
- c)No, it is not linear because it does not satisfy the vector addition property.
- d)Yes, it is linear
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Is the transformation T(x, y, z) = (x, y, 0) linear?a)No, it is not li...
We are given that a transformation defined by
T(x,y,z) = (x,y, 0)
Let α and β be scalars and (x1, y1, z1) and (x2, y2, z2) be two vectors. Then
T[α(x1, y1, z1) + β(x1, y2, z2)]
or equivalently
Hence, T is linear.
T(x,y,z) = (x,y, 0)
Let α and β be scalars and (x1, y1, z1) and (x2, y2, z2) be two vectors. Then
T[α(x1, y1, z1) + β(x1, y2, z2)]
or equivalentlyHence, T is linear.
Most Upvoted Answer
Is the transformation T(x, y, z) = (x, y, 0) linear?a)No, it is not li...
Explanation:
To determine whether the transformation T(x, y, z) = (x, y, 0) is linear, we need to check if it satisfies two properties: scalar multiplication and vector addition.
Scalar Multiplication Property:
For a transformation to be linear, it must satisfy the property that T(c * v) = c * T(v), where c is a scalar and v is a vector.
Let's consider an arbitrary vector v = (x, y, z) and a scalar c. Applying the transformation T to cv, we have T(cv) = T(cx, cy, cz) = (cx, cy, 0).
Now, let's compute c * T(v). We have c * T(v) = c * T(x, y, z) = c * (x, y, 0) = (cx, cy, 0).
Since T(cv) = c * T(v) for all vectors v and scalars c, the scalar multiplication property is satisfied.
Vector Addition Property:
For a transformation to be linear, it must satisfy the property that T(u + v) = T(u) + T(v), where u and v are vectors.
Let's consider two arbitrary vectors u = (x1, y1, z1) and v = (x2, y2, z2). Applying the transformation T to u + v, we have T(u + v) = T((x1 + x2, y1 + y2, z1 + z2)) = (x1 + x2, y1 + y2, 0).
Now, let's compute T(u) + T(v). We have T(u) + T(v) = T(x1, y1, z1) + T(x2, y2, z2) = (x1, y1, 0) + (x2, y2, 0) = (x1 + x2, y1 + y2, 0).
Since T(u + v) = T(u) + T(v) for all vectors u and v, the vector addition property is satisfied.
Conclusion:
Since the transformation T(x, y, z) = (x, y, 0) satisfies both the scalar multiplication and vector addition properties, it is a linear transformation. Therefore, the correct answer is option D: Yes, it is linear.
To determine whether the transformation T(x, y, z) = (x, y, 0) is linear, we need to check if it satisfies two properties: scalar multiplication and vector addition.
Scalar Multiplication Property:
For a transformation to be linear, it must satisfy the property that T(c * v) = c * T(v), where c is a scalar and v is a vector.
Let's consider an arbitrary vector v = (x, y, z) and a scalar c. Applying the transformation T to cv, we have T(cv) = T(cx, cy, cz) = (cx, cy, 0).
Now, let's compute c * T(v). We have c * T(v) = c * T(x, y, z) = c * (x, y, 0) = (cx, cy, 0).
Since T(cv) = c * T(v) for all vectors v and scalars c, the scalar multiplication property is satisfied.
Vector Addition Property:
For a transformation to be linear, it must satisfy the property that T(u + v) = T(u) + T(v), where u and v are vectors.
Let's consider two arbitrary vectors u = (x1, y1, z1) and v = (x2, y2, z2). Applying the transformation T to u + v, we have T(u + v) = T((x1 + x2, y1 + y2, z1 + z2)) = (x1 + x2, y1 + y2, 0).
Now, let's compute T(u) + T(v). We have T(u) + T(v) = T(x1, y1, z1) + T(x2, y2, z2) = (x1, y1, 0) + (x2, y2, 0) = (x1 + x2, y1 + y2, 0).
Since T(u + v) = T(u) + T(v) for all vectors u and v, the vector addition property is satisfied.
Conclusion:
Since the transformation T(x, y, z) = (x, y, 0) satisfies both the scalar multiplication and vector addition properties, it is a linear transformation. Therefore, the correct answer is option D: Yes, it is linear.
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Is the transformation T(x, y, z) = (x, y, 0) linear?a)No, it is not linear because all z components map to 0b)No, it is not linear because it does not satisfy the scalar multiplication property,c)No, it is not linear because it does not satisfy the vector addition property.d)Yes, it is linearCorrect answer is option 'D'. Can you explain this answer?
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