Consider The Mapping T : R² → R Such That T((x₁, x₂)) = (x₁, x₂). a. Show That T Is A Linear Transformation. b.
In the realm of linear algebra, understanding the properties of mappings between vector spaces is fundamental. The mapping given here, T : R² → R, defined by T((x₁, x₂)) = (x₁, x₂), appears straightforward but warrants a detailed examination to determine whether it is a linear transformation. This article explores this mapping comprehensively, beginning with the precise definition of the transformation, followed by a rigorous proof of linearity, including the verification of two critical properties: additivity and scalar multiplication. This exploration not only elucidates the characteristics of T but also reinforces the foundational concepts underlying linear transformations.
Understanding the Mapping T
Definition of the Mapping
The given mapping T operates from the vector space R², which consists of all ordered pairs of real numbers, to R. The definition is explicitly given as:
- For any vector (x₁, x₂) in R², T((x₁, x₂)) = (x₁, x₂).
At first glance, this resembles the identity map on R², but the key point here is that the codomain is R rather than R². Given the expression, it appears that T maps from R² to R, but the notation suggests a possible typo or ambiguity: perhaps the intended mapping is from R² to R², or perhaps the target space is R. For clarity, we proceed assuming the mapping is from R² to R², with T((x₁, x₂)) = (x₁, x₂), i.e., the identity transformation. If the intended target is R, then T would have to produce a scalar, but since the formula produces a pair, the most consistent interpretation is that T maps R² to R², with T((x₁, x₂)) = (x₁, x₂). For the purpose of this article, we assume the latter, as it aligns with the notation and the structure of the transformation.
Clarifying the Transformation
Assuming T : R² → R² with T((x₁, x₂)) = (x₁, x₂), it is essentially the identity map on R². The identity transformation is a fundamental example in linear algebra, serving as a baseline for understanding more complex mappings. To analyze whether T is a linear transformation, we need to verify the two defining properties of linearity:
- Additivity: T(u + v) = T(u) + T(v) for all u, v in R².
- Homogeneity (scalar multiplication): T(cu) = cT(u) for all u in R² and all scalars c in R.
Part A: Showing T Is a Linear Transformation
Step 1: Verify Additivity
Let u = (u₁, u₂) and v = (v₁, v₂) be arbitrary vectors in R². Their sum is:
- u + v = (u₁ + v₁, u₂ + v₂)
Applying T to u + v:
T(u + v) = T((u₁ + v₁, u₂ + v₂)) = (u₁ + v₁, u₂ + v₂)
Applying T separately to u and v and then adding:
T(u) + T(v) = (u₁, u₂) + (v₁, v₂) = (u₁ + v₁, u₂ + v₂)
Since T(u + v) = T(u) + T(v), the additivity property holds.
Step 2: Verify Homogeneity (Scalar Multiplication)
Let c be an arbitrary scalar in R, and u = (u₁, u₂) be an arbitrary vector in R². Then:
- cu = (cu₁, cu₂)
Applying T to cu:
T(cu) = T((cu₁, cu₂)) = (cu₁, cu₂)
On the other hand, c times T(u) is:
c T(u) = c (u₁, u₂) = (c u₁, c u₂)
Since T(cu) = c T(u), the homogeneity property is satisfied.
Conclusion of Part A
Because T satisfies both additivity and homogeneity, we conclude that the mapping T : R² → R², defined by T((x₁, x₂)) = (x₁, x₂), is a linear transformation. This is consistent with the fact that the identity map preserves all linear structure and is a prototypical example of a linear transformation.
Part B: Additional Properties and Implications
Identity Transformation and Its Significance
The transformation T described here is essentially the identity map on R², which is a fundamental linear transformation. Its properties include:
- Preservation of vector addition and scalar multiplication
- Having the identity matrix as its standard matrix representation
- Being invertible, with its inverse also being the identity map
Matrix Representation of T
Since T is the identity map, its matrix representation with respect to the standard basis is the 2×2 identity matrix:
[1 0] [0 1]
This matrix acts on vectors in R² via standard matrix-vector multiplication, confirming the linearity of T.
Implications of Linearity
Understanding that T is linear has several consequences:
- It simplifies the analysis of T's behavior on arbitrary vectors
- It allows us to leverage matrix algebra to study transformations
- It confirms that T is a structure-preserving map between vector spaces
Summary and Final Remarks
In this article, we thoroughly examined the mapping T : R² → R² defined by T((x₁, x₂)) = (x₁, x₂). By verifying the fundamental properties of additivity and scalar homogeneity, we demonstrated that T is indeed a linear transformation. Recognized as the identity map, T exemplifies the quintessential linear transformation that preserves the structure of the vector space R². Understanding such mappings is crucial in the broader context of linear algebra, as they serve as building blocks for more complex transformations and applications across mathematics and engineering fields.
