Question #79918
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
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
Expert's answer
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Comments
Thank you for correcting us. The answer in part ii) is correct only when T is one-to-one and the range of T is equal to W. If it is not true, then the statement of ii) may be false.
In part 2, how you wrote that V and W have same dimension?Take T:R²->R³ s.t T(x,y)=(x,y,0) which is one one and not invertible also R² and R³ are of finite dimensions.