Determine which of the following mappings are linear mappings.
A. The mapping L defined by L(x_1, x_2 ) = (4x_1- 2x_2, 3 |x_2|).
B. The mapping L defined by L(x_1, x_2 ) = (2x_1-3x_2, x_1+4, 5x_2).
C. The mapping L defined by L(x_1, x_2, x_3 ) =(x_1, 0 , x_3).
D. The mapping L defined by L(x_1, x_2, x_3 ) =(1, x_2, x_3).
E. The mapping L defined by L(x_1, x_2, x_3 ) =(x_1, x_2, -x_3).
To determine whether a mapping is linear, we need to check if it satisfies two properties: additive and homogeneous.
A mapping is considered additive if L(u + v) = L(u) + L(v) for all vectors u and v in the domain.
A mapping is considered homogeneous if L(cu) = cL(u) for any scalar c and vector u.
Let's go through each option:
A. The mapping L defined by L(x_1, x_2) = (4x_1 - 2x_2, 3 |x_2|).
To check the additive property:
L(u + v) = L(x_1 + x_1', x_2 + x_2') = (4(x_1 + x_1') - 2(x_2 + x_2'), 3 |x_2 + x_2'|)
= (4x_1 - 2x_2, 3 |x_2|) + (4x_1' - 2x_2', 3 |x_2'|)
= L(x_1, x_2) + L(x_1', x_2')
Therefore, this mapping is additive.
To check the homogeneous property:
L(cu) = L(cx_1, cx_2) = (4(cx_1) - 2(cx_2), 3 |cx_2|)
= c(4x_1 - 2x_2, 3 |x_2|)
= cL(x_1, x_2)
Therefore, this mapping is homogeneous.
Since mapping A satisfies both the additive and homogeneous properties, it is a linear mapping.
B. The mapping L defined by L(x_1, x_2) = (2x_1 - 3x_2, x_1 + 4, 5x_2).
To check the additive property:
L(u + v) = L(x_1 + x_1', x_2 + x_2') = (2(x_1 + x_1') - 3(x_2 + x_2'), (x_1 + x_1') + 4, 5(x_2 + x_2'))
= (2x_1 - 3x_2, x_1 + 4, 5x_2) + (2x_1' - 3x_2', x_1' + 4, 5x_2')
This does not equal L(x_1, x_2) + L(x_1', x_2'), so this mapping is not additive.
Therefore, this mapping is not linear.
C. The mapping L defined by L(x_1, x_2, x_3) = (x_1, 0, x_3).
To check the additive property:
L(u + v) = L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (x_1 + x_1', 0, x_3 + x_3')
= (x_1, 0, x_3) + (x_1', 0, x_3')
This equals L(x_1, x_2, x_3) + L(x_1', x_2', x_3'), so this mapping is additive.
To check the homogeneous property:
L(cu) = L(cx_1, cx_2, cx_3) = (cx_1, 0, cx_3)
= c(x_1, 0, x_3)
= cL(x_1, x_2, x_3)
Therefore, this mapping is homogeneous.
Since mapping C satisfies both the additive and homogeneous properties, it is a linear mapping.
D. The mapping L defined by L(x_1, x_2, x_3) = (1, x_2, x_3).
To check the additive property:
L(u + v) = L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (1, (x_2 + x_2'), (x_3 + x_3'))
= (1, x_2, x_3) + (1, x_2', x_3')
This equals L(x_1, x_2, x_3) + L(x_1', x_2', x_3'), so this mapping is additive.
To check the homogeneous property:
L(cu) = L(cx_1, cx_2, cx_3) = (1, cx_2, cx_3)
= c(1, x_2, x_3)
= cL(x_1, x_2, x_3)
Therefore, this mapping is homogeneous.
Since mapping D satisfies both the additive and homogeneous properties, it is a linear mapping.
E. The mapping L defined by L(x_1, x_2, x_3) = (x_1, x_2, -x_3).
To check the additive property:
L(u + v) = L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (x_1 + x_1', x_2 + x_2', -(x_3 + x_3'))
= (x_1, x_2, -x_3) + (x_1', x_2', -x_3')
This does not equal L(x_1, x_2, x_3) + L(x_1', x_2', x_3'), so this mapping is not additive.
Therefore, this mapping is not linear.
To summarize:
- Mapping A is a linear mapping.
- Mapping B, C, and D are linear mappings.
- Mapping E is not a linear mapping.
To determine whether a mapping is linear, we need to check if it satisfies two conditions: additive property and scalar multiplication property.
1. Additive property: A linear mapping satisfies the equation L(u+v) = L(u) + L(v), for any vectors u and v.
2. Scalar multiplication property: A linear mapping satisfies the equation L(cu) = cL(u), for any scalar c and vector u.
Let's go through each mapping and check if they satisfy these properties:
A. The mapping L defined by L(x_1, x_2 ) = (4x_1- 2x_2, 3 |x_2|).
To check the additive property:
L(x_1 + x_1', x_2 + x_2') = (4(x_1 + x_1') - 2(x_2 + x_2'), 3 |x_2 + x_2'|)
= (4x_1 - 2x_2, 3 |x_2|) + (4x_1' - 2x_2', 3 |x_2'|)
= L(x_1, x_2) + L(x_1', x_2')
Thus, the mapping L satisfies the additive property.
To check the scalar multiplication property:
L(c x_1, c x_2) = (4(c x_1) - 2(c x_2), 3 |c x_2|)
= c(4x_1 - 2x_2, 3 |x_2|)
= cL(x_1, x_2)
Thus, the mapping L satisfies the scalar multiplication property.
Therefore, mapping A is linear.
B. The mapping L defined by L(x_1, x_2 ) = (2x_1-3x_2, x_1+4, 5x_2).
To check the additive property:
L(x_1 + x_1', x_2 + x_2') = (2(x_1 + x_1') - 3(x_2 + x_2'), (x_1 + x_1') + 4, 5(x_2 + x_2'))
= (2x_1 - 3x_2, x_1 + 4, 5x_2) + (2x_1' - 3x_2', x_1' + 4, 5x_2')
= L(x_1, x_2) + L(x_1', x_2')
Thus, the mapping L satisfies the additive property.
To check the scalar multiplication property:
L(c x_1, c x_2) = (2(c x_1) - 3(c x_2), (c x_1) + 4, 5(c x_2))
= c(2x_1 - 3x_2, x_1 + 4, 5x_2)
= cL(x_1, x_2)
Thus, the mapping L satisfies the scalar multiplication property.
Therefore, mapping B is linear.
C. The mapping L defined by L(x_1, x_2, x_3 ) =(x_1, 0 , x_3).
To check the additive property:
L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (x_1 + x_1', 0, x_3 + x_3')
= (x_1, 0, x_3) + (x_1', 0, x_3')
= L(x_1, x_2, x_3) + L(x_1', x_2', x_3')
Thus, the mapping L satisfies the additive property.
To check the scalar multiplication property:
L(c x_1, c x_2, c x_3) = (c x_1, 0, c x_3)
= c(x_1, 0, x_3)
= cL(x_1, x_2, x_3)
Thus, the mapping L satisfies the scalar multiplication property.
Therefore, mapping C is linear.
D. The mapping L defined by L(x_1, x_2, x_3 ) =(1, x_2, x_3).
To check the additive property:
L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (1, x_2 + x_2', x_3 + x_3')
= (1, x_2, x_3) + (0, x_2', x_3')
= L(x_1, x_2, x_3) + L(x_1', x_2', x_3')
Thus, the mapping L satisfies the additive property.
To check the scalar multiplication property:
L(c x_1, c x_2, c x_3) = (1, c x_2, c x_3)
= c(1, x_2, x_3)
= cL(x_1, x_2, x_3)
Thus, the mapping L satisfies the scalar multiplication property.
Therefore, mapping D is linear.
E. The mapping L defined by L(x_1, x_2, x_3 ) =(x_1, x_2, -x_3).
To check the additive property:
L(x_1 + x_1', x_2 + x_2', x_3 + x_3') = (x_1 + x_1', x_2 + x_2', -(x_3 + x_3'))
= (x_1, x_2, -x_3) + (x_1', x_2', -x_3')
= L(x_1, x_2, x_3) + L(x_1', x_2', x_3')
Thus, the mapping L satisfies the additive property.
To check the scalar multiplication property:
L(c x_1, c x_2, c x_3) = (c x_1, c x_2, -c x_3)
= c(x_1, x_2, -x_3)
= cL(x_1, x_2, x_3)
Thus, the mapping L satisfies the scalar multiplication property.
Therefore, mapping E is linear.
In summary, the linear mappings are:
A. The mapping L defined by L(x_1, x_2) = (4x_1 - 2x_2, 3 |x_2|)
B. The mapping L defined by L(x_1, x_2) = (2x_1 - 3x_2, x_1 + 4, 5x_2)
C. The mapping L defined by L(x_1, x_2, x_3) = (x_1, 0, x_3)
D. The mapping L defined by L(x_1, x_2, x_3) = (1, x_2, x_3)
E. The mapping L defined by L(x_1, x_2, x_3) = (x_1, x_2, -x_3)