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Why am I getting wrong solution to the system $2x+6y-3z=10$, $5x+2y-1z=12$?

\begin{align} 2x+6y-3z&=10 \tag 1 \\ 5x+2y-1z&=12 \tag 2 \end{align} Matrix form: $AX=b$ $A=\begin{pmatrix} 2 & 6&-3\\5&2&-1\end{pmatrix}$ $X=\begin{pmatrix}x\\y\\z\end{pmatrix}...
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Question about change-of-basis matrix

One can view this process in two different ways: $1).\ $ using the definition of a change of basis matrix, one has that $U,$ viewed as a linear isomorphism of $V$, satisfies $Ue_i=e'_i.$ That is all ...
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Is there a variant of the dot-product operation that returns $\frac{a_1}{b_1} + \frac{a_2}{b_2}$ from vectors $[a_1,a_2]$ and $[b_1, b_2]$?

To answer the asked question: NO. While there are many inverse simplification formulas that exist, none can be resolved to fulfill your requested outcome for all input values. I agree that the ...
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Diagonalizable matrices over $\mathbb{C}$

I didn't check if your polynomial $p$ is actually the characteristic polynomial. Assuming it is, its derivative has roots $-2$ and $4/3$. Since $p(-2)>0$ and $p(4/3)<0$, it follows that $p$ has ...
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Diagonalizable matrices over $\mathbb{C}$

If $p=\chi_A$ is reducible over $\mathbb{Q}$, then it has a rational root (because of degree $3<2+2$). By Gauß lemma, it will then have an integral root. Since $p$ is monic, this integral root $\xi$...
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  • 1,562
0 votes

Is it true that $A\cap (A\cap B)^{\perp} = A\cap B^{\perp}$?

The question is equivalent to: Does $x\in A$ and $x$ orthogonal to $A\cap B$ already imply $x$ orthogonal to $B$? Suppose $y$ in $B$ so that $x$ is not orthogonal to $y$. This can only be the case if $...
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5 votes
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The determinant of a matrix is "linear in the rows "

The definition of the determinant is a sum of sign weighted products of matrix elements (it seems from the top of the included image). Show that in each of these products exactly one element of row $l$...
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  • 1,562
1 vote

How to prove ${\lambda _n}\left( {{X^T}AX} \right) \le {\lambda _n}\left( A \right){\rho ^2}\left( X \right)$?

We may assume that $\lambda_n(X^TAX)>0,$ as otherwise the inequality is obvious. Thus $X$ and $A$ are invertible. We have $$\sigma(X^TAX)=\sigma(XX^TA)=\sigma(A^{1/2}XX^TA^{1/2})$$ where $\sigma(S)$...
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Are differential operators in differential equations additive?

Differentiation is a linear operator from which $$\frac{d}{dx}\left(f(x)+g(x)\right)= \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\tag{1}$$ $$\frac{d}{dx}(c \cdot f(x))= c\cdot \frac{d}{dx}(f(x))\tag{2}$$ ...
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  • 9,864
2 votes

Are differential operators in differential equations additive?

Yes, differentiation is a linear operator. Also, from the definition we can see this: $$ \frac{d}{dx}[A(x) + B(x)] = \lim_{h \to 0} \frac{A(x+h)+B(x+h) - A(x)-B(x)}{h} = $$ $$\lim_{h \to 0} \frac{A(x+...
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  • 3,433
1 vote

Are differential operators in differential equations additive?

The question title doesn't really match up with what you're asking. The sum of two differentiable functions is differentiable. On the other hand, two differential equations can be added in the ...
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1 vote

Are differential operators in differential equations additive?

Very much additive ! That Property enables us to get Derivative of $x^3+2x^2+3x+4$ as the Sum of the Derivatives of $x^3$ (which is $3x^2$) & $2x^2$ (which is 4x) & $3x$ (which is 3) & $4$ ...
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  • 853
3 votes

Is there an orthonormal set of polynomials whose derivatives are orthogonal?

The simplest example is provided by the Chebyshev polynomials of the first and the second kind. They are orthogonal on $[-1,1]$ with respect to the weights $(1-x^2)^{-1/2}$ and $(1-x^2)^{1/2}$ ...
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0 votes

Is there an orthonormal set of polynomials whose derivatives are orthogonal?

Looking at the references in the paper mentioned by 3mOo, we find a paper by Krall with the following result: theorem . If $\{\phi_n(x)\}$ is a set of orthogonal polynomials with the weight function $...
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  • 2,682
0 votes

Is there an orthonormal set of polynomials whose derivatives are orthogonal?

I think we might be able to prove this in the negative via a few facts about orthogonal polynomials. This is too much for a comment so I am going to start and answer that I might try to finish later ...
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2 votes

Is there an orthonormal set of polynomials whose derivatives are orthogonal?

Maybe this could be useful: https://digitalcommons.unl.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1032&context=mathfacpub in particular the Theorem (page 880) that states the ...
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  • 535
0 votes

Why can't I estimate the Fundamental Matrix from a coplanar set of points?

In the book, Three-dimensional computer vision by Olivier Faugeras (Page 271) is explained why you can't have coplanar points. Basically it's because the matrix $kron(x,x^\prime)$ has to be of rank ...
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Determinant of Kroneker product, approach via $F[x]$-modules

Three comments. You don't need to know anything about modules to compute that if $v, w$ are eigenvectors of two linear maps $f, g$ with eigenvalues $\lambda, \mu$ then $(f \otimes g)(v \otimes w) = f(...
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Given $x\in \mathbb{R}^n$, Is there exist a matrix $A\in \mathbb{R}^{n\times n}$ such that $A$ only have $x$ as the null space?

Assume $x\in\Bbb{R^n}\setminus \{0\}$ ( otherwise it's trivial) Let $x=(x_1, x_2, \ldots, x_n) $ and suppose $x_i\neq 0$ for some $1\le i\le n$ Then consider the matrix $A\in M_n(\Bbb{R})$ such that $\...
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  • 9,910
0 votes

Similar problem of Hadamard's maximal determinant problem

Actually Hadamard's determinant inequality is more general and would apply to your question. The determinant of any $n\times n$ complex matrix $A$ with columns $a_i$ obeys $$ | \det(A) |\leq \prod_{1\...
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  • 6,626
7 votes
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Given $x\in \mathbb{R}^n$, Is there exist a matrix $A\in \mathbb{R}^{n\times n}$ such that $A$ only have $x$ as the null space?

If $x = 0$, it suffices to take an invertible matrix $A$ (such as the identity matrix). So, I will consider only the case where $x \neq 0$. One convenient way to construct a suitable matrix is to use ...
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2 votes

Given $x\in \mathbb{R}^n$, Is there exist a matrix $A\in \mathbb{R}^{n\times n}$ such that $A$ only have $x$ as the null space?

For $x=0$ just take the identity matrix. For $x\neq 0$ take the vector $x$ and make it part of a basis $\mathcal{B}=\{x,v_2,…,v_n\}$. Take the matrix $$B=\begin{bmatrix}0&0\\0&I_{n-1}\end{...
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0 votes

Given $x\in \mathbb{R}^n$, Is there exist a matrix $A\in \mathbb{R}^{n\times n}$ such that $A$ only have $x$ as the null space?

Yes. One way to generate such an A is as follows: Take the first column of an $n\times n$-matrix $T$ to be $x$ and choose arbitrary $n-1$ other vectors for the remaining columns such that they form a ...
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  • 1,562
1 vote
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Find the result of T given matrices against ordered basises

There are two ways that I can think of that you can solve the question. $1$. Simply reconstruct $T$ using the columns of the matrix. Consider the following example on how to do that. Example: Consider ...
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  • 1,122
1 vote

Find the result of T given matrices against ordered basises

It seems like there is an error in the question. If $B$ is the matrix of the transformation $T$ with respect to the standard basis, then the matrix with respect to the ordered basis $(v_{1},v_{2})$ ...
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  • 684
3 votes

Is a basis of $V$ a basis of $V/U$?

The answer is, as you hint at, no unless $U=0$. The set $\left\{v_{1}+U,...,v_{n}+U\right\}$ is a generating set for $V/U$, but generally not linearly independent. Consider the following example. Put ...
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  • 684
2 votes
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Is a basis of $V$ a basis of $V/U$?

$v_1+U,\dots,v_n+U$ spans $V/U$ but it is not linearly indeprndent. For example, $v_1+U=v_2+U$ if $v_1-v_2 \in U$. Specific example: Let $V=\mathbb R^{2},U=\{(x,x): x \in \mathbb R\}$. Let $v_1=(1,0),...
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1 vote

Substitution in a matrix

It is because $a=-a \iff a=0$.
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  • 478
1 vote

$A^k$ and nonzero elements

The proposed bound is not true. Let $n=2m$ and $$a_{ij}=\begin{cases} 0 &j\le m\\ 0 & j>m,\ i>m\\ 1 & j>m,\ i\le m \end{cases}$$ Then $A^2=0.$ The matrix $A$ has $m^2=n^2/4$ ...
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2 votes
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A $2$x$2$ Matrix $A$ is invertible if $\det(A) \not= 0$

Note that $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = (ad-bc) I$.
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1 vote
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Exercise $8$, Section $3.D$ - Linear Algebra Done Right

No, it is not correct. You claim that, since $T$ is surjective, $\dim\operatorname{null}T=0$. That is false. Consider, for instance, $T\colon\Bbb R^2\longrightarrow\Bbb R$ defined by $T(x,y)=x$. It is ...
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3 votes
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How to translate a 3d point along a line?

Let $a,b$ be the two vectors in question. We can define a third vector $d=b-a$. This vector will point from $a$ to $b$. Normalize this to the unit vector $n$: $$n=\frac{d}{|d|}$$ Now, if you have some ...
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  • 3,433
2 votes

How to translate a 3d point along a line?

Any point $M$ on the line going through $A$ and $B$ can be represented as $$M = A + t\cdot \overrightarrow{AB}$$ where $t$ is a real number. For instance, $t=0$ gets you $M=A$, while $t=1$ gets you $M=...
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2 votes
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Is this vector inside the box formed by 3 other vectors?

Taking Wolfram MathWorld at its word (since I was somewhat uncertain, but finding it made me feel a lot better): A vector $v \in \mathbb{R}^3$ is in the parallelipiped made by $x_1,x_2,x_3 \in \mathbb{...
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1 vote
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Determinant of Kroneker product, approach via $F[x]$-modules

You need to tensor over $F$, not over $F[x]$ to get the Kronecker product. The action of $x$ on $V_f\otimes_F V_g$ is defined on elementary tensors as $x(v \otimes w):=xv \otimes xw$ and extended ...
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0 votes

Exercise 11, Section 3.5 of Hoffman’s Linear Algebra

Since $W_1\cap W_2$ is a subspace of $W_2$, there is a complementary subspace $U_2$ such that $W_2=(W_1\cap W_2)\oplus U_2$. Let $U$ be a complement of $W_1+W_2$ in $V$, So $U\oplus U$ is a complement ...
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Symmetric matrix in $\mathbb{Z}_2$

We proceed by induction on $n$. We use the inductive hypothesis to its fullest potential: If I remove row $i$ and column $i$, then the statement is true by IH. Hence there exists $x_1, \cdots, x_{i-1},...
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  • 442
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Example of reciprocal equation of type II of even degree

There are four types of reciprocal equation $f(x)=0:$ i) Type 1 - even degree(in this case of degree 4) -$a_0$ and $a_4$ are same sign - solution divide the equation by $x^2$ for degree $4$ and $x^3$ ...
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Let $A,B$ be matrices under certain conditions, then $AB=B$

Let us leave out each Matrix in G and take only the Index of the Matrix. That is $G=\{1,2,3,4,5,6,7,8,\cdots,k\}$ Let $B=(1,2,3,4,5,6,7,8,\cdots,k)$ which is a sequence of Integers. Sum of elements of ...
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  • 853
3 votes
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Let $A,B$ be matrices under certain conditions, then $AB=B$

We don't actually need the fact that the elements are matrices at all. Instead of the ring of matrices, $G$ can be a subset of any ring $R$ (not necessarily commutative) if $G$ is a finite group using ...
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  • 5,444
0 votes

Direct sum of kernels of factors of a polynomial annihilating $A$.

Consider the polynomials $$Q_j(x)={ P(x)\over (x-c_j)^{m_j}}$$ The polynomials $Q_1,Q_2,\ldots, Q_k$ are relatively prime. Therefore there exist polynomials $R_j$ such that $$R_1Q_1+R_2Q_2+\ldots +...
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1 vote

Cauchy-Schwarz-like inequality for the wedge product

I had a similar question, and found a general bound here (p. 9, equation 1.10) given by $$ \|\mathbf{a} \wedge \mathbf{b}\| \le \begin{pmatrix} k + \ell \\ k \end{pmatrix}^{1/2} \|\mathbf{a}\|\|\...
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1 vote

Let $A,B$ be matrices under certain conditions, then $AB=B$

Notice that the map $f_{i}:G\to G$ defined by $f_i(g)=A_ig$ is a bijection for every $i$.
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  • 1,564
1 vote

If we know the direction of the sum of vectors $\frac{\sum_1^\infty w_i}{\|\sum_1^\infty w_i\|}=u$, how can we calculate the expection of the vectos?

I am not certain if this answer is correct, but I believe it gets to what OP wanted, based on the OP? I think I understand what is being asked here, so let me reveal what I believe the question ...
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4 votes
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Quadratic form defined by a permutation of projection matrix

The result you're looking for holds as a consequence of the Cauchy Schwarz inequality. I will use $\|x\|$ to refer to the norm $\|x\| = \sqrt{x'x}$. We note that $\|Px\| = \|x\|$ for all $x$. Thus, we ...
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An inequality on trace of product of two matrices

Let's write $A = \sum_{i=1}^{n}\lambda_i u_iu_i^T$ and $B = \sum_{i=1}^{n}\mu_i v_iv_i^T$. $$AB = \sum_{i,j} \lambda_i \mu_j u_i^Tv_j u_iv_j^T$$ and so $$\text{Tr}(AB) = \sum_{i=1}^n \sum_{j=1}^n \...
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2 votes

If we know the direction of the sum of vectors $\frac{\sum_1^\infty w_i}{\|\sum_1^\infty w_i\|}=u$, how can we calculate the expection of the vectos?

To put the question rigorously, assume you have $W$ a random vector. $\left(W_n\right)_{n\ge1}$ are independent observations of $W$. If $\mathbb E\left\|W\right\|^2 < \infty$ then by the law of ...
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1 vote
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A upper triangular matrix with zeros in the diagonal is nilpotent

Let $V=V_1=F\times F\times F$, $V_2=\{0\}\times F\times F$, and $V_3=\{0\}\times\{0\}\times F$, and $V_4$ to be the zero subspace. Notice that $V_4\subseteq V_3\subseteq V_2\subseteq V_1$. One way to ...
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  • 143k
1 vote

Hyperplane in a complex vector space

Since this has gotten bumped, and may be useful to posterity: Literally, a complex hyperplane in a finite-dimensional complex vector space is (i) a real subspace of real codimension two that (ii) is ...
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1 vote

Determinant of symmetric matrix

It is quite straightforward to arrive at a triangular matrix with the same determinant: After the following elementary row operations: $$R_1\to R_1-R_2\\ R_2\to R_2-R_3\\ R_3\to R_3-R_4\\ R_4\to R_4-...
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