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revised Prove that $\vdash p \lor \lnot p$ is true using natural deduction
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Dec
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comment Friend B and C have eaten zero apples. How many more apples has C eaten?
haha good point @user46944
Dec
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asked Friend B and C have eaten zero apples. How many more apples has C eaten?
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answered Spectral theorem for matrices
Nov
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comment Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
In what kind of book could I learn about why matrix multiplication, $Q\mapsto Q^{-1}$ is continuous, and $Q\mapsto Q^\top$ is continuous? I own Topology by James R. Munkres, would that be a good one? @GEdgar
Nov
21
comment Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
@mathcounterexamples.net First things that come to mind is that $f^{-1}(A)$ is open for every open subset $A$ of $O(n)$.
Nov
21
comment Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
Then every entry is a polynomial of $q_{11}, ... , q_{nn}$ and therefore continuous.
Nov
21
comment Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
@GEdgar It is some time ago, since I studied this. But something things are coming back. Could/should I see it as a mapping $f_A: \Bbb R^{n\times n} \to \Bbb R^{n\times n}$ because then it makes more sense to me. With the euclidean metric.
Nov
21
comment Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
@GEdgar I have surely proved that a couple of years ago in a real analysis course. But with matrices, I get a bit confused to be honest.
Nov
21
asked Given $A$ a symmetric square matrix. Why is $f_A: O(n) \to \mathbb{R}^{n \times n}: Q \mapsto Q^{\top} A Q$ continuous?
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