Find the Pre-Image A=[[4],[-2],[6]] , [[6],[-3],[9]]

Problem

A={[[4],[−2],[6]],[[6],[−3],[9]]}

Solution

1. Identify the task as finding the pre-image of a set of vectors, which typically refers to the set of all vectors v such that a linear transformation T(v) maps to the given set A

2. Observe the relationship between the vectors in the set A

3. Factor out common scalars from each vector to see if they are linearly dependent.

4. Simplify the first vector:

[[4],[−2],[6]]=2*[[2],[−1],[3]]

5. Simplify the second vector:

[[6],[−3],[9]]=3*[[2],[−1],[3]]

6. Conclude that both vectors lie on the same line in ℝ^3 passing through the origin and the vector [[2,−1,3]]^T Without a specific transformation T provided, the pre-image cannot be calculated numerically; however, the set A represents a subset of the span of a single basis vector.

Final Answer

A={2*[[2],[−1],[3]],3*[[2],[−1],[3]]}

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