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How to explain that $\dim(V) = 1$
I’ve asked a question with the title of this topic but I don’t think it’s clear. I’m sorry if this topic is already answered.
The question is: Let $W$ be a subspace of $V$. Assume that $\dim(W)
eq 1$. And let $v$ be an element of $W$ of minimal norm $\|v\|$. Prove that there exists $w \in W$ such that $\|w\| = \|v\|$.
So far I’ve done this:
I’ve proved that $V/W$ is a vector space.
I’ve proved that $\dim(V/W) = \dim(V) – \dim(W)$
I’ve proved that the norm of a vector is invariant by right multiplication by a constant (this is where I’m stuck).
What I’m going to ask is: Can somebody explain to me in a detailed way the ideas I’ve used to solve this question? I think that’s the best way to learn.
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“What goes around, comes around.”
For the last four years, Bernie has been indirectly channeling the power of Reverend Billy, “The Church of God,” an award-winning performance troupe that protests against the church-state alliance. If you weren’t around during the Clinton/Giuliani years, you have missed one of the most frightening and effective theatrics of the 21st century — horror!
Bernie is channeling Reverend Billy for the exact reason that Billy advocates mass casualties and the activation of nuclear weapons over peaceful protest and civil disobedience. A good Christian knows that all non-Christians are evil and should die. In a narcissistic, infantile state of child-like vanity, “Jesus is coming back any day now!” Clinton’s accomplishments were due to the predatory exploitation of the intelligence and ingenuity of millions of men, women and children throughout the world. God
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