Linear Transformation

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Linear Transportation/Transmission

Definition

• Linear transportation and homomorphism are the same thing.
• Assume U and V are two vector spaces over field F. In this situation, F(U,V) is called linear transmission or homomorphism.
• Properties:
  • F(α + β) = F(α) + F(β)
  • F(cα) = cF(α) where α and β come from U and c is a scalar.
• The image mapping from U to V is called homomorphic image.

Example

• Questions are asked whether the given mapping is linear transportation or not.
• Different concepts are explained using various methods.

Kernel and Null Space

• Kernel: All elements that map to the identity (such as 0).
• Null Space: It's a subset of the kernel. Its dimension is called nullity.

Rank and Nullity

• The dimension of the final dimension of a vector space is called rank.
• Nullity is the dimension of the null space.
• Final dimension = Rank + Nullity.

Linear Transformation Proof

Example 1: Vector Spaces giving A, B, and C

• When cα and dβ are taken, it is shown that F(α + β) = F(α) + F(β).
• If it does not show constants alone, like x², y², then it will not be linear transportation.

Matrix Representation

• Representing V3R defined by basics such as 100, 010, 001.
• Converting transformations like 2b+c and 3a-4b into matrices.

Important Formulas

• T(U + V) = T(U) + T(V)
• T(cα) = cT(α)

Key Ideas

• Calculating linear transformation using rank and nullity.
• Understanding kernel, null space, and their dimensions.
• Using correct options and methods to save time.
• The importance of linear transformation from the perspective of various exams.

In the End

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