# Tagged Questions

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### Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
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### Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
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### “The Conjugate of a matrix”

I am having some trouble understanding a definition/question in my linear algebra text book. The question states " If $A$ is a square matrix, a matrix of the form $P^{-1}AP$ where $P$ is invertible ...
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### How to prove the uniqueness of linear functional

$\textbf{Theorem}$ If $V$ is a $n$-dimensional vector space, if $\{x_1,.,.,., x_n\}$ is a basis in $V$ and if $\{\alpha_1,\cdots \alpha_n\}$ is any set of $n$ scalars, then there is one and only one ...
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### Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
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### Doubt in proof of Dual of the direct sum

If $M$ and $N$ are subspaces of $V$, and if $V = M \oplus N$, then $$V' = M^\perp \oplus N^\perp$$ where $W^\perp$ is the annihilator of $W$. I didn't understand how to prove both of the ...
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### Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
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### Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...