# Tagged Questions

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### Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
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### Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
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### Writing up a rigorous solution for finding a basis for the $n \times n$ symmetric matrices.

I am aware that a similar question has been asked here, among other questions, but I feel that my question is different because I am actually trying to write up a very rigorous proof that such a set ...
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### Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
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### Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
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### Proof: $ker(g) \subset ker(f)$ ..

Let $V = \mathbb{R}_{\le 3} [x]$ with basis $B = (1, x, x^2, x^3)$. And $f: V \to \mathbb{R}, p \to \int_{-1}^1 p(x) dx$ and $g: V \to \mathbb{R}^3, p \to ^t( p(-1), p(0), p(1) )$. (1) I had to ...
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### Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
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### Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
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### If two matrices are similar, the geometric multiplicities of their eigenvalues are the same

Problem Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of ...
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### Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
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### Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
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### Proving Direct Sum

Claim. Let $V$ be a vector space over $F$, and suppose that $W_1$, $W_2$, and $W_3$ are subspaces of $V$ such that $W_1 + W_3 = W_2 + W_3$. Then $W_1 = W_2$. I know that this claim is false, but ...
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### Showing something is not onto?

Quick question..: If I have a linear transformation $T:\mathbb R^2 \to \mathbb R^4$, can I say that since $\mathbb R^4$ is greater than $\mathbb R^2$, it can never be onto - since the elements in ...
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### proof $B^{-1}MB$ is triangular

How to prove this? Theorem: Let $M$ be a matrix of complex numbers. There exists a non-singular matrix $B$ such that $B^{-1}MB$ is a triangular matrix. This is corollary from book Linear Algebra by ...
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### Linear Algebra Proof with one-dimensional subspaces

Suppose that V is finite dimensional, with $dimV=n$. Prove that there exist one-dimensional subspaces $U_1,...,U_n$ of $V$ such that $$V = U_1 \oplus\dotsb\oplus U_n$$ My linear algebra is rusty, very ...
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### Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
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### Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
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### Proof that the set of all invertible $n \times n$ matrices of real numbers is a vector space.

I'm studying Algebra and I'm asked to prove or disprove the statement above. I found that is true, but I'm not sure how to prove it. My problem here is that this statement is too "broad", i.e. I ...
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### Given $A$ and $\vec{b}$ in $A\vec{x}=\vec{b}$, solve for $\vec{x}$

What are the steps to solve for $\vec{x}$, given that $A\vec{x}=\vec{b}$ and we know what $A$ and $\vec{b}$ are? I know the first thing you do is multiply each side by $A^T$ ...
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### Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
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### Linear algebra hw! Linear transformation

Let $T : V -> V$ be a linear transformation where V is a nite dimensional vector space. If rank(T) = rank$(T^2)$, prove that image(T)$\cap$Ker(T) = {0}. I have to give this hw to my prof this ...
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### Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
I need to prove something in my homework I just don't know how to approach it and need some guidance. "Show that for a matrix $A$ ($n \times m$) and a vector $\vec{x}$ ($m \times 1$) it applies that: ...
### Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.
Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along $Y$. Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$. I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are ...