Questions tagged [linear-algebra]
For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
121,352
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Proof verification: $[L_1L_2:K] \leq [L_1:K][L_2:K]$
Let $K \subseteq L_1,L_2 \subseteq L$ be fields. I want to show the following:
$$[L_1L_2:K] \leq [L_1:K][L_2:K]$$
My attempt:
The definition of $$L_1L_2:=L_1(L_2):=\bigcap \{F:F\text{ is a subfield ...
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1
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1
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14
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Partial isometry, Isometry and Gram matrix
Suppose I have a set of matrices $\mathcal{H}$ s.t. $\forall H\in\mathcal{H}, \ H = CU$ where $C$ is a fixed matrix and $U$ is an orthonormal matrix. We know that this set can be characterized ...
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How does multiplying a row and a col with a constant affect semi-positive definite property?
Sorry if this question is obvious....
Assuming $A$ is semi-positive definite (and symmetric). Now, if we want to multiply all elements on the $i$-th row AND all elements on the $i$-th column of $A$ by ...
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Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial (x−1)^2 (x−2) [closed]
Is this true. Because I can't find any invertible P, such that M = P D P^-1, so that the similarity satisfies. Can any one guide me regarding my approach to the problem?
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11
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Exponentials of stochastic band matrices
This may be a duplicate, but I've done some searching and I can't find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.
I'm ...
- 1,460
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3
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39
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Find orthogonal vectors in relation to span
Consider $R^3$ as an inner product space in relation to scalarmultiplication. Find all vectors in the subspace
$$Span\bigg(\bigg[\begin{matrix}1\\2\\1
\end{matrix}\bigg],\bigg[\begin{matrix}3\\4\\1
\...
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How to determine if a subspace of $ \mathbb{R}^n $ has an integer basis
Let $ W $ be a sub vector space of $ \mathbb{R}^n $. How can we determine if $ W $ admits an integer basis?
This is equivalent to asking how to determine if $ W \cap \mathbb{Z}^n $ spans $ W $.
- 5,622
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2
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63
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What is a sufficient and necessary condition for a system of linear equations to have a unique integer solution?
Bézout's theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $\gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution ...
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Determine a system of linear equations whose solution set is the affine hull of the following points:
$(2, 1, −1, 1)^T$, $(−3, −1, 2, 0)^T$
I don't know if I'm doing this right:
2$x_1$+$x_2$-$x_3$+$x_4$=3
-3$x_1$-$x_2$+2$x_3$=-2
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0
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25
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Name of symmetric multilinear function?
What's the name (if any) for the multilinear symmetric function that maps a quadratic matrix over some commutative ring $R$ with $1$ to a value in $R$? It's basically like the determinant, but ...
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28
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Find an orthogonal basis to minimize the norm
Let $A \in R^{n\times n}$. Find O, an orthogonal basis of $R^{n\times n}$, to minimize $∥A−O∥_F$.
$∥⋅∥_F$ is the Frobenius Norm.
It is similar to this question, except that O is not an orthogonal ...
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1
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47
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using computer to solve a system of nonlinear equations
I was wondering if there is a program or website that solved a system of nonlinear equations
I was trying a system that looks like this
$$a+b = cd , a+c = bd ,a+d = bc$$
, ...
it's like a pattern
that ...
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17
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Determinant of involving adjacency matrix of subgraph
Let $A$ be the adjacency matrix of a graph $G$ labelled $\{x_1,\cdots,x_n\}$. Let $B:=(I-\frac{1}{2d}A)^{-1}$ where $d$ is a positive integer.
Then let $A'$ be the adjacency matrix of the graph $G\...
- 1,437
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$u_1, u_2$ is linearly independent if and only if the family $u_1 + u_2, u_1 − u_2$ is linearly independent.
I have this exercise where I want to check my solutions. Can someone help me?
Let $u_1$ and $u_2$ be elements of $V$. If $1 + 1 ≠ 0$ in $K$, then the family $u_1, u_2$ is linearly independent if and ...
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Finding the kernel of a specific linear transformation
Let $k$ be a field, $M:k^n\to M_{r,s}(k)$ be a linear map (where $M_{r,s}(k)$ denotes the set of $r\times s$ matrices with coefficients in $k$), let $N:k^n\to M_{s,s}(k)$ be a linear map, and let $p\...
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Show that there exists exactly one $l_A$-invariant $K$-subspace of dimension $1$ in $K^2$.
So I have this exercise where I want to check my solutions. Can someone help me?
Let $A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K)$ with $a, b, d ∈ K$, where $a ≠ 0$ and $d ≠ ...
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Homogeneous equations for conic sections
The equation of a "standard" circle is $x^2 + y^2 = r^2$. That equation is not homogeneous and does not include the origin; we can homogenize it by adding a $z$ term of degree 2, getting $x^...
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Left operator on $M_n(\mathbb{F})$ diagnolizable iff A is diagnolizable
Question: let $A\in V = M_n(\mathbb{F})$,left translation operator on V is defined as $T:V\rightarrow V,B\rightarrow T(B)=AB$, prove that $T$ is if and if only A is diagnolizable.
I have found a ...
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If we expand the definition of the general quadratic $ax^{2}+bx+c=0$ to include the case $a=0$, can we arrive at a general solution?
Through a simple mathematical substitution, I have stumbled upon an alternative formula for solving a quadratic equation:
$$x=\frac{2c}{-b \pm \sqrt{b^{2}-4ac}}$$
(Please refer to my formula ...
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Where is the gap in the proof of uniqueness of the row reduced echelon form?
The author of the book I'm reading says that the following proof has a gap,
but I don't know where it is.
Here is the proof which the author claims that it has a gap.
First, The author defines that '...
- 1,129
5
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1
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62
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The Definition of Orthogonal Complement
In Linear Algebra Done Right by Axler, the author defines the orthogonal complement as follows:
If $U$ is a subset of a vector space $V$, then the orthogonal complement of $U$, denoted by $U^\perp$, ...
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0
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39
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Visualization of a standard two dimensional polyhedron
I am reading this book on linear programming, and the authors give an excellent exposition of the topic by the interplay between the underlying algebra and geometry. Their main approach is to motivate ...
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2
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28
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Show $2m(R\textbf{x})\cdot \textbf{a} +m\|R\textbf{a}\|^2=m\|R\textbf{x}\|^2+m\|R(\textbf{a}-\textbf{x}) \|^2$
I need to show that
$$2m(R\textbf{x})\cdot \textbf{a} +m\|R\textbf{a}\|^2=m\|R\textbf{x}\|^2+m\|R(\textbf{a}-\textbf{x})
\|^2$$
Here $R=I-P$ where $P=UU^T$, and $\textbf{a},\textbf{x}\in\mathbb{R}^m$, ...
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1
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Using the Hessian to determine convexity of $f(x,y) = \ln(x + y)$
Disclaimer: This is a homework question that I'm currently working on.
I have been asked to use the Hessian to determine if $f(y_1, y_2) = \ln(y_1 + y_2)$ is convex, concave, or neither.
I've computed ...
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Interior of a set defined by continuous non-strict inequalities is the set with strict inequalities
How to prove the following?
The interior of a set $$\left\{ x \mid f_i(x) \leq 0, \forall i \right\}$$ where $f_i$ are continuous, is $$\left\{ x \mid f_i(x) < 0, \forall i \right\}$$
Frequently, ...
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What is the result of a elementary $k$ alternating tensor acting on $k$ different vectors?
I just learn the concept of elementary k form: on an open set $A$. That is if $x\in A$ , then we have $$\begin{align*}
\phi_I(x)=\phi_{i_1}(x) \wedge \cdots \wedge \phi_{i_k}(x)
\end{align*}$$ ...
- 1,237
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$f(m+n)+f(mn)=f(m)f(n)+1$ [closed]
How to find all functions $f:R\rightarrow R$ such that
$f(m+n)+f(mn)=f(m)f(n)+1$
I I've tried a lot, but I didn't find a solution
I only know that $f(x)=1 $ and $f(n)=n+1$ are solutions
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Step by step derivation of vector cross-product expression
Equation 1
$$\vec{a} * \vec{b} = (|\vec{a}| | \vec{b}| sin θ)ň $$
Equation 2
$$\vec{a} * \vec{b} = (a_yb_z - a_zb_y)î - (a_xb_z - a_zb_x)ĵ + (a_xb_y - a_yb_x)k̂ $$
I have managed to figure out and ...
0
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1
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64
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Do convergent power series form a vector space? [closed]
In my linear algebra class, I was wondering if convergent power series form a vector space?
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1
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59
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If A and B are positive semidefinite matrices, then $\mathrm{tr}( A^3B + AB^3)\ge 2\mathrm{tr}( A^2B^2)$. [closed]
We know the following relationship: If A and B are positive semidefinite matrices, then $\mathrm{tr}(AB)^2\le\mathrm{tr}(A^2B^2)$.
How can I prove this relationship?
If A and B are positive ...
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Is vector space of all matrices with nonzero determinant a vector space? [closed]
I was wondering if vector space of all matrices with nonzero determinant a vector space?
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1
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25
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Finding a representation of the inverse as linear combination
I am struggling with the following:
Write $(7+2^{\frac{1}{3}})^{-1} \in \mathbb{Q}(2^{\frac{1}{3}})$ as a $\mathbb{Q}$-Linear combination of $\{1,2^{\frac{1}{3}},2^{\frac{2}{3}}\}$. Hint: This means ...
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Alternatives to Hermite Normal Form that preserve positivity of variables..
I have an application that generates directed acyclic multi-graphs. From these I generate path-edge adjacency matrices. Paths are rows and edges are the columns. I am exploring the column space ...
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Finding the basis of all perpendicular vectors to two vectors in R4... How many vectors should this basis include?
I have been given this question in my textbook:
Find a basis of the subspace of R4 that consists of all
vectors perpendicular to both
<1, 0, -1, 1> and <0, 1, 2, 3>
My question is how many ...
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How to solve minimisation using dual and simplex method
How would I minimise $2y_1 + y_2$ using the simplex method?
subject to:
$ 10y1 + y2 \ge 10 $
$ 2y1 + y2 \ge 8 $
$ y1 + y2 \ge 6 $
$ y1 + 2y_2 \ge 10 $
$ y1 + 12y_2 \ge 12 $
$ y1,y2 \ge 0 $
I have got ...
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30
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Linear Algebra 2nd Edition Robert Messer
Does anyone know where I can find solutions to Robert Messer's "Linear Algebra: Gateway to Mathematics"? I bought the book online, but the back only has selected solutions to the practice ...
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10
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How does accuracy of eigenvalues from lanczos algorithm change with tridiagonal matrix size
Trying to implement Lanczos algorithm, to find the lowest $M$ eigenvalues of a very large $N$ by $N$ matrix $H$.
I have implemented a working algorithm, which lets me find matrix $V$ with orthonormal ...
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How can I transform nxm matrix A into nxm matrix B?
I have two numerical matrices of size 48x50, $A$ and $B$. I'd like to obtain the transformation of $A$ into $B$ and use the same transformation to transform $A'$ into $B'$. I don't know how to ...
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If $f(a v) = a^2 f(v) $ and $B (v,w) := f(v+w) - f(v) - f(w) $is a bilinear form then $f$ is a quadratic form.
I read the following statement in Conway's book on quadratic forms and tried to prove it:
A function $f$ is a quadratic form if and only if $f(a\textbf {v}) = a^2 f(\textbf{v}) $ and $B (\textbf{v},\...
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What is the motivation behind this proof, any list of vector with length n+1 in the span of an independent list of vector with length n is dependent?
Although I can understand each step of the proof, the process seems random to be and I do not understand why in each step, the author choose to do the manipulation this way. I have read the proof for ...
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paint a board with two colors without repeating the amount painted in each row rows
A child is playing coloring his chessboard and will paint each square either completely blue or completely red. To give it a personalized touch, he wants to paint the same number of red squares as ...
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0
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15
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$Ax = y$ and $A'x = y'$ have the same solution set $L$, then $L = ∅$ or $\ker(l_A) = \ker(l_{A'} )$
Hey I have problems solving this exercise. Can someone help me?
Let $A, A' ∈ Mat_{m,n}(K)$ and $y, y' ∈ K^m$.
Let $l_A: K^n → K^m$ and $l_{A'} : K^n → K^m$ be the $K$-linear maps associated with $A$ ...
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0
answers
7
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How to compute MAD of a dataset in arbitrary directions?
This is how variance in the direction $d$ is computed for a dataset $X = \left\{ x_1, x_2, \cdots, x_N \,|\, x_i \in \mathbb{R}^M \right\}$, where $\sum^N_i x_i = 0$:
$$ \sigma^2(d) = \sum^N_i \left( ...
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0
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23
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Representing matrix of $id_V , f$ and $f^2$
Hey I have this exercise and I want to see if what I have done is correct.
Let $v_1$ and $v_2$ be a basis of the vector space $V$. Let $f: V \rightarrow V$ be the $K$-linear function with $f(v_1) = ...
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0
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32
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Spectrum of $(A^TA-B^TB)$ and $(AA^T-BB^T)$
I am wondering whether one can say anything useful about the relationship of $A^TA-B^TB$ and $AA^T-BB^T$ in terms of their spectra?
In particular I know that their traces are the same since $tr(A^TA-B^...
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votes
0
answers
15
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Showing that all matrices of a linear operator have the same rank. [closed]
Can someone help me with proof?
Show that all matrices of a linear operator have the same rank.
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1
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1
answer
28
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Suppose $V=\Bbb R_2[x]$ and $W=\{p(x) \in V: p(1)=0\}$. Define a subspace $U=\{T:V \to V : W \subseteq \operatorname{Ker} T\}$, find $\dim U$
Suppose $V=\Bbb R_2[x]$ (all polynomials of degree $2$ or less), and $W=\{p(x) \in V: p(1)=0\}$. Define a subspace $U=\{T:V \to V : W \subseteq \operatorname{ker} T\}$. Find $\dim U$.
I know $W$ is ...
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0
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52
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Consultation regarding my Undergraduate Thesis [closed]
I would like to ask if I can study an undergraduate thesis regarding adjacency matrix of a special types of digraphs. I want to study their special properties if there is a relation between its ...
0
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0
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36
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Proving a statement on linear algebra involving inequalities, determinants, and eigenvalues.
Suppose $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,...
- 4,722
1
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1
answer
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polynomial representation of determinant
The question I'm working on is, given a matrix $A = (a_{i,j})_{i,j = 1}^n$ and the polynomial
\begin{gather*}
P(x) : = \det\begin{bmatrix}a_{1,1} + x & a_{1,2} + x & ... & a_{1,n} + x\\...
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