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Linear Algebra Done Right

Exercise 2.C.1

Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$ such that dim $U =$ dim $V$ . Prove that $U = V$ .

Answers

Let $u_{1}, u_{2}, . . ., u_{m}$ be the basis of $U$ .

Because it is a basis, it obviously is linearly independent. Additionally, since dim $U =$ dim $V$ , the length is the same as the basis of $V$ .

Therefore, $u_{1}, . . ., u_{m}$ is also a basis of $V$ , showing that $U$ = $V$ .

Arden_Shi

2025-03-18 17:52

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