Question

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) R2 has infinitely many non-zero, proper vector subspaces.
ii) If T : V → W is a one-one linear transformation between two finite dimensional vector spaces V andW then T is invertible.
iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.
iv) Every unitary operator is invertible.
v) Every system of homogeneous linear equations has a non-zero solution.

02 Feb 2018
Answer :
Word Count : 429

Let's go through each statement and determine whether it is true or false, providing a justification for each:

i) R^2 has infinitely many non-zero, proper vector subspaces.
TRUE. 
Justification: In R^2, you can find infinitely many non-zero, proper vector subspaces. For example, you can take any line passing through the origin in R^2, and it will be a proper vector subspace. There are infinitely many such lines, so there are infinitely many non-zero, proper vector subspaces.

ii) If T: V → W is a one-one (injective) linear transformation between two finite-dimensional vector spaces V and W, then T is invertible.
TRUE.
Justification: If T is injective, it means that for every element in V, there is a unique element in W that maps to it under T. This unique mapping allows us to define an inverse transformation, T^(-1): W → V, such that T^(-1)(T(v)) = v for all v in V. This makes T invertible.

iii) If A^k = 0 for a square matrix A, then all the

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