# Tagged Questions

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### Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
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### Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
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### Proving $v$, $T(v)$, $T^2(v)$ is a basis

I'm trying to prove the following: Given that $V$ is a vector space, with $dim V = 3$, and $T: V \to V$ is a linear map with the properties $T^2(v) \neq 0$ and $T^3(v) = 0$, with $v \in V$, show that ...
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### Can you check my proof concerning an invariant subspace under a diagonilizable linear operator and its complementary invariant subspace?

This was an exercise problem from H&K Linear Algebra(sec 7.2, exercise 18). Could you check my proof? The theorem is as follows: If $V$ is a finite-dimensional vector space and $W$ is an ...
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### Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
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### Proof Verification, Uniqueness of vector $v$ satisfying $\varphi(u) = \langle u,v\rangle$ for a linear functional $\varphi$.

I want to prove the uniqueness of the following vector $v$. The existence of the vector is guaranteed. We know that there exists at least one vector $v$ for every $u$ such that for a linear ...
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### VERIFYING: Proving an $n \times n$ Matrix Vector Space

Let $V$ be the vector space made by the $n \times n$ square matrices. a) Prove that $S=\{A\in V|A^t=A\}$ is a subspace of $V$ b) Prove that $T=\{A\in V|A^t=-A\}$ is a subspace of $V$ c) Prove that ...
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### Prove $\{(x_1 ,x_2, 0) : x_1, x_2 ∈ F\}$ is a subspace of $F^3$.

$(x_1 ,x_2, 0) + (y_1, y_2, 0) =((x_1 + y_1), (x_2 + y_2), 0)).$ So, it's closed under +. $a(x_1 ,x_2, 0) = ax_1, ax_2, 0$. So, it's closed under *. Vector $(0, 0, 0) \in \mathbb F^3$ and its ...
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### Find matrices complying to given constraints

We are given linear mapping of $n$-dimensional vector space, such as: It has $n+1$ eigenvectors Any $n$ of them are linearly independent Find all matrices which could define such a linear ...
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### Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...
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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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### Showing AB=0 does not imply either A,B=0, but that singular

Ex. 8.5 - Mathematical Methods for Physics and Engineering (Riley) By considering the matrices  A = \left( \begin{matrix} 1 & 0 \\ 0 & 0 \\ \end{matrix} \right) ...
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### Why is this proof complete if only one condition is satisfied?

In the 10th ed. of Elementary Linear Algebra (Anton), the following statement exists: If Ax = b is a system of linear equations, exactly one of the following is true: (a) the system has no ...
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### Find an ordered basis of $V$ such that $[T]_\beta$ is a diagonal matrix.

The entire problem statement is: Let $V$ be a finite dimensional vector space and $T:V\to V$ be the projection of $W$ along $W'$, where $W$ and $W'$ are subspaces of $V$. Find an ordered basis ...
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### A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
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### Strategy of a purely algebraic proof of Cayley-Hamilton Theorem

Let $p(\lambda)=det(A-\lambda l)$ be the characteristic polynomial of a $n \times n$ matrix $A$. Then $p(A)=O.$ Let $p(\lambda)=p_{0}+p_{1}\lambda+\ldots+p_{n-1}\lambda^{n-1}+p_{n}\lambda^{n}$. ...
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### What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
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### Why must P be orthonormal, and not just orthogonal, for change of variable in Quadratic Form? [Kolman P560 8.8.24]

Lay P402 : A change of variable is an equation of the form $x=Py$, where $P$ is an invertible matrix and $y$ is the (neW) coordinate vector of $x$ relative to the basis of $\mathbb{R}^{n}$ determined ...
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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 ...
### Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim
Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...