# Tagged Questions

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### Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$(A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I.$$ I know that $A^{T}A$ = $I$, but where can I go ...
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### Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
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### Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
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### Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
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### Finding matrices values

I was trying to teach myself some things about saddle points. This is a little more advance when it comes to finding the number for each matrices and a value $v$ $($say $v=(1/3), (1/3), (1/3))$ ...
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### Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
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### Prove that if product of matrices is singular, one of the matrices is singular.

I'm having trouble with this proof, it would be much easier to work out the other way it seems. Let $A$ and $B$ be square matrices of equal size. Prove that if $\det(AB) = 0 =C$ then either $A$ or ...
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### Question about $e^T$ where T is a transformation

First off, I'm given a matrix $A$ s.t. the characteristic polynomial of $A$ is $p(a) = (-1)^nx^n+x^2-x+2$ and am asked to find $det(A^k)$ for a natural $k$ and $det(e^A)$. So from the polynomial I get ...
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### Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
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### Integral Problem $\sin^6 x$. [closed]

What is the integral of $\sin^6 x$? Can some one publish it with the method it would be really helpful. Specially I want the trig identities.
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### Proof: Show there is set of $n+1$ points in $\mathbb{R}^n$ such that distance between any two distinct points is $1$?

Argh, I hate to ask a question again so soon, especially one I feel like I should know. Linear algebra is taking its toll, and I am not quite used to the theory side of mathematics. Anyways, I ...
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### Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
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### Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
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### Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
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### Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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### Proving $x_i = \dfrac{\det(A_i)}{\det(A)}$

Suppose we have an nxn matrix A, and $A\underline{x} = \underline{p}$ prove that $x_i = \dfrac{\det(A_i)}{\det(A)}$ where $A_i$ is the matrix obtained from A by replacing the $i^{th}$ column by ...
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### Why is it sufficient for a normal to only be orthogonal to 2 vectors on a plane instead of 3?

The following is an excerpt from my textbook: It is clear geometrically that there is a unique plane containing any 3 points $A,B$ and $C$ that are not all on a line. In determining the equation ...
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### Prove the identity

$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8} = \frac{\sin x}{ 8\sin \frac{x}{8}}$$ Conjecture a generalization of this result and prove its correctness by induction. Ps: I have ...
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### Endomorphism- Nilpotent matrices

An endomorphism $f: V \rightarrow V$ of an $F$-vector space is called nilpotent iff there exists $\delta \in \mathbb N$ such that $f^\delta=0$. Suppose that $f : V\rightarrow V$ is a nilpotent ...
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### How can we prove that an affine function can be written as $f(x)=ax+b$?

Sorry if duplicate. Searched but could not find anything. I understand the concept of an affine function. I am not sure as to how to go about writing a proof for the definition $f(x)=ax+b$ where ...
Assume that $(4k + 3) ^ 2 - (4k + 3)$ is not divisible by 4. If this is true, prove that $(4(k+1) + 3) ^ 2 - (4(k+1) + 3)$ is not divisible by 4. I need to prove this for my induction problem, and ...
These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n}$ is a real ...