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.

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42 views

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 ...
1 vote
1 answer
14 views

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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1 answer
10 views

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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-1 votes
0 answers
21 views

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?
0 votes
0 answers
11 views

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 ...
0 votes
3 answers
39 views

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 \...
1 vote
0 answers
25 views

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 $.
0 votes
2 answers
63 views

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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0 answers
9 views

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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25 views

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 ...
0 votes
0 answers
28 views

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 answer
47 views

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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0 answers
17 views

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\...
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1 vote
2 answers
48 views

$u_1, u_2$ 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 ...
0 votes
0 answers
25 views

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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0 answers
6 views

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 ≠ ...
2 votes
0 answers
42 views

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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3 votes
1 answer
45 views

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 ...
5 votes
1 answer
103 views

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 ...
1 vote
1 answer
33 views

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 '...
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5 votes
1 answer
62 views

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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1 vote
0 answers
39 views

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 ...
0 votes
2 answers
28 views

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 $\textbf{a},\textbf{x}\in\mathbb{R}^m$, ...
1 vote
1 answer
34 views

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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-1 votes
0 answers
37 views

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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1 vote
1 answer
29 views

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 $$\begin{align*} \phi_I(x)=\phi_{i_1}(x) \wedge \cdots \wedge \phi_{i_k}(x) \end{align*}$$ ...
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4 votes
0 answers
97 views

$f(m+n)+f(mn)=f(m)f(n)+1$ [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
1 vote
1 answer
72 views

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 votes
1 answer
64 views

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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0 votes
1 answer
59 views

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 ...
-3 votes
0 answers
62 views

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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0 votes
1 answer
25 views

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 ...
0 votes
0 answers
11 views

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 ...
1 vote
2 answers
38 views

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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0 votes
0 answers
18 views

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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0 votes
0 answers
30 views

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 ...
0 votes
0 answers
10 views

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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-2 votes
0 answers
54 views

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 ...
1 vote
1 answer
38 views

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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0 votes
0 answers
45 views

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 ...
1 vote
1 answer
39 views

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 ...
-1 votes
0 answers
15 views

$Ax = y$ 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 $A$ ...
0 votes
0 answers
7 views

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 votes
0 answers
23 views

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) = ...
1 vote
0 answers
32 views

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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-3 votes
0 answers
15 views

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 vote
1 answer
28 views

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 ...
-1 votes
0 answers
52 views

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 votes
0 answers
36 views

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,...
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1 vote
1 answer
41 views

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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