Let V be the vector space of all 2*3 real matrices and W be the vector space of all 2 * 2 real matrices. Thena)There is a one-one linear transformation from V to Wb)Kernel of any linear transformation from V to W is non-trivialc)There is an isomorphism from V to Wd)There is an onto linear transformation from W to VCorrect answer is option 'B'. Can you explain this answer?

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Answer

Understanding the Vector Spaces V and W
- Vector Space V: This consists of all 2x3 real matrices. The dimension of V is 6 because each matrix entry can be independently chosen from the real numbers (2 rows × 3 columns = 6).
- Vector Space W: This consists of all 2x2 real matrices. The dimension of W is 4 (2 rows × 2 columns = 4).
Analyzing the Options
- a) One-one linear transformation from V to W:
- A linear transformation from V (dimension 6) to W (dimension 4) cannot be one-to-one. This is because the dimension of the kernel must be at least 2 (since 6 > 4), implying it cannot be injective.
- b) Kernel of any linear transformation from V to W is non-trivial:
- True. Given that the dimension of V is greater than that of W, any linear transformation from V to W must have a kernel of dimension at least 2. Therefore, the kernel is non-trivial, meaning it contains more than just the zero vector.
- c) Isomorphism from V to W:
- An isomorphism requires a one-to-one and onto mapping. Since V and W have different dimensions (6 and 4, respectively), no isomorphism exists.
- d) Onto linear transformation from W to V:
- While it is possible to have an onto mapping from W to V, this statement does not contradict the conclusion about the kernel being non-trivial in option b.
Conclusion
The correct answer is option 'b' because any linear transformation from the higher-dimensional space V to the lower-dimensional space W will necessarily have a non-trivial kernel, confirming that it cannot be injective. Thus, the kernel contains more than just the zero vector, making this statement valid.

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