# Tag Info

## New answers tagged vector-spaces

1 vote

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• 10k
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### Different versions of Courantâ€“Fischer theorem

If we replace $A$ by $-A$ and $k$ by $n-k+1$ in the statement of the min-max version of the theorem, we get \begin{align*} \lambda_{n-k+1}(-A) &=\min_{\dim S=n-k+1}\,\max_{x\in S\setminus0}\frac{x^...
• 140k
1 vote
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### Closure of projecting cone is the tangent cone

Proof of $T(Y, \bar{y}) \subseteq \operatorname{cl} P(Y-\{\bar{y}\})$: If $h \in T(Y, \bar{y})$, then there exists a sequence of nonnegative reals $t_l$ and points $y_l \in Y$ converging to $\bar{y}$ ...
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### $ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric.

This proof looks correct! Indeed, for an matrix $M$, if $M^T M$, then for any vector $x$, $$\lVert Mx\rVert^2 = x^T M^T M x = 0,$$ from which any $Mx = 0$ and so $M = 0$. In terms of a concise ...
• 4,062
1 vote
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### $ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric.

Note that for real matrices $M$, $M = 0$ if and only if $M^TM = 0$. More generally, for a positive semidefinite matrix $P$, it holds that $M^TPM = 0$ if and only if $\sqrt{P}M = 0$, which holds if and ...
• 226k
1 vote

• 750
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### Why is $R(A^*) \perp N(A)$ true?

Notation: By $\langle x,y\rangle$ I mean the standard inner product on $\Bbb{C}^{n}$ given by $y^{*}x$. Let $x\in R(A^{*})$ . Then $x=A^{*}(z)$ for some $z\in \Bbb{C}^{n}$ and consider some $y\in N(A)$...
• 14.3k
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### Why is $R(A^*) \perp N(A)$ true?

If $u \in N(A)$, then $0 = (Au, v) = (u, A^*v)$ for all $v$, so $u \in R(A^*)^{\perp}$. Hence $N(A) \subset R(A^*)^{\perp}$. Since $\dim N(A) = \dim R(A^*)^{\perp}$, we get $N(A) = R(A^*)^{\perp}$.
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### Is $\ell^1$ a closed subspace of the normed space $(\ell^2, \||\cdot\||_2)$?

No, all closed subspaces of a Hilbert space are isomorphic to Hilbert space. $\ell^1$ is not isomorphic to Hilbert space. The meaning of the sentence: "This converges in $\ell^2$ norm to the full ...
• 18.6k
1 vote
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### How is sphere related with Rayleigh quotient?

The key point is to see it as inner products. We have for $x\in\mathbb{R}^n\backslash \{ 0\}$ : $$R_A(x) =\frac{\langle x, Ax\rangle} {\langle x, x\rangle} =\frac{\langle x, Ax\rangle} {||x||^2 }.$$ ...
• 470
1 vote

### If $B$ is a $n \times n$ matrix, then is the column space of $B$ and $BB^{'}$ same?

This works even if $B$ is $m \times n$. We just need to show that, for any $x \in \Bbb{R}^n$, there exists some $y \in \Bbb{R}^m$ such that $Bx = BB'y$. To construct such a $y$, we take a tip from ...
• 51.2k
1 vote
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### Completeness of Y (normed tvs) in the proof of th. 2 in ch. V section 3 of Yosida's Functional Analysis ed 6

My first attempt, as noted by @Jochen, was flawed. Here is a (hopefully) correct solution: Let us denote by $\tau$ the topology on $Y$ induced by the weak topology on $X$ and by $\tau_1$ the norm ...
• 843
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### How orthogonal projection connects with eigen space?

The matrix $P=\begin{bmatrix}0&0\\a&1\end{bmatrix}$ is a projection since $P^2=P$ but it is not orthogonal unless $a=0$ since it is not symmetric. The nullspace of this matrix also depends on ...
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### Is this a vector space: $k(x,y,z)=(kx,y,z)$?

Your space fails to be a vector space because your multiplication does not distribute over addition: $$(a+b)\cdot(x,y,z)=((a+b)x,y,z)=(ax+bx,y,z)$$ a(x,y,z) + b(x,y,z)= (ax,y,z)+(bx,y,z)=(ax+bx,y+y,...
Accepted

### Is this a vector space: $k(x,y,z)=(kx,y,z)$?

You are correct in saying that $V$ is not a vector space, but your friend is right in saying that your argument is wrong. Indeed, within the axiom (which would be more correctly written as) For each ...
• 226k