Linear algebra is concerned with 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 ...

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Orthogonal rewriting of multi-vectors

Use Orthogonal rewriting of multi-vectors to verify $(a \land b \land c)$$(d \land e)$ where a,b,c,d,e are vectors in a dimension of at least 5? Note: by definition the grade $\vert r-s \vert$ vector ...
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Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
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What is the meaning of cofactor expansion?

I understand how to perform a cofactor expansion in finding the determinant. Can you explain what this method is really capturing or what thinking leads us to use this method?
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18 views

How do I calculate similar matrix with arbitrary change of base matrix

$P=[\begin{matrix}\overrightarrow v_1 & \overrightarrow v_2 & \overrightarrow v_3\end{matrix}]$ with $\overrightarrow v_1$, $\overrightarrow v_2$, $\overrightarrow v_3 \in \Bbb R^3$ and $A= \...
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33 views

Prove that sum of 2 bivectors in $R^{4}$ is not necessarily a bivector

Prove that sum of 2 bivectors in $R^{4}$ and other higher dimensional spaces is not necessarily a bivector and show that $B^{2}$ is not a scalar? I want to show that $B$ does not have a specific ...
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Is there an algorithm for finding the largest possible linear subspace of a given vector space having this specific property?

Let $G_1,G_2,\dots,G_k$ be $n\times n$ real matrices, and let $\mathcal{G} = \operatorname{span}\left\{ G_k\right\}$. Let $\mathcal{V}$ be a linear subspace of $\mathcal{G}$, i.e. $\mathcal{V} \...
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Prove that the sum of 2 bivectors in $R^{3}$ is a bivector? Hint:Think geometrically

$R^{3}$ is noted as a 3 dimensional space. I know that pseudo-scalars are bivectors on a plane. also I know bivectors have more freedom than scalars and vectors on a plane. Can I use $a \land b$ + $...
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What are the constraints on $\alpha$ so that $AX=B$ has a solution?

I found the following problem and I'm a little confused. Consider $$A= \left( \begin{array}{ccc} 3 & 2 & -1 & 5 \\ 1 & -1 & 2 & 2\\ 0 & 5 & 7 & \alpha \end{...
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34 views

linearly dependent family of vectors.

Can someone help me to solve this question please : Establish, by induction, that : $ \forall n \in \mathbb{N} \setminus \{ 0,1 \} \ \forall v_1 , \dots , v_n \in \mathbb{R}^n $ linearly independants ...
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28 views

silver bronze linear optimization

A non-profit offers crafts complimentary gift packages for its donors. The non-profit costs for each package are \$4 for the Bronze level package, \$7 for the Silver level package, and \$9 for the ...
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33 views

Matrix that changes basis

Does change of basis matrix we use in linear transformations change both the domain and range of the transformation matrix? By the way, I have a hard time calculating the change of basis matrix. I've ...
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4answers
45 views

Show that the diagonal elements are not all $0$

If the rank of a real symmetric matrix be $1$, show that the diagonal elements of the matrix can not be all zero. Since the rank is $1$, the determinant of the entire matrix is $0$, so it is ...
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41 views

represent an image in linear algebra

Can we represent a grayscale image as a matrix of values, and then apply all our linear algebra techniques to that? Like finding the column space and null space, reason about the matrix structure. ...
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29 views

Finding Curvatures of Parabolas

Use the formula for the center of curvature c of a curve $r=r(t)$ at a arbitrary point r(t) given by $c-r=-\dot r^{3}(\dot r \land \ddot r)^{-1}$ to find the center of curvature of the semicubical ...
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45 views

When do $k$ vectors span $\mathbb{R}^n$? ($k>n$)

My specific question I'm having trouble with is finding the values of $a$ for which $v_1=(1,3,4), v_2=(2,-1,1), v_3=(-3,5,a^2-2), v_4=(4,2,a+4)$ span $\mathbb{R}^3$. I'm relatively new to Linear ...
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18 views

Number of Subspaces that contains other Space

In $GF(2)$, How Can I calculate the number of subspaces of dimension $k<w$ that contains a fixed subspace of dimension $k'<w$:
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40 views

Intuitive way to understand the use of matrix inversion to find dual basis

I'm currently thinking about the following problem: Problem: Let $B = (b_1, b_2, b_3)$ a base of $\mathbb{R}^3$. Find the correlating dual basis $B^* = (b_1^*, b_2^*, b_3^*)$. $B$ is explicitly ...
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nilpotent linear transformation and invariant subspaces

I'm trying to proof a biconditional statement about a nilpotent linear transformation, and I think I already proved it one way,but I'm stuck on the other way. The statement is as follows: Let $\...
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11 views

How many linearly independent solutions particular case over $GF(2)$

Let $H$ a matrix with dimensiones $r\times n$, where its entries are indeterminates in $GF(2)$. Let $Y\in GF(2)^r$ a random vector. My question How Can I determine the maximum number of sets of ...
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1answer
58 views

How can I divide a vector by a matrix?

I am trying to go backwards through a neural network. I have an output and I want to see what input would lead to that output. To go forwards I start with a vector and multiply by a matrix and then ...
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25 views

Is there a non-trivial special orthogonal transform which preserves the diagonal elements of a symmetric matrix with positive entries?

This problem is at the interface of matrix algebra and spectral graph theory. Let $\mathbf{S}$ be a symmetric $n\times n$ matrix, with positive entries $S_{ij}\geq 0$, and $\mathbf{D} = \mathrm{diag}(...
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2answers
32 views

Matrix decomposition in unipotent matrices

Consider the positive definite and symmetric matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 6 & -1 \\ 0 & -1 & 1 \end{pmatrix}$$ Find a decomposition with unipotent $U ...
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Typo in Hoffman and Kunze's linear algebra book

I'm reading Hoffman and Kunze's linear algebra book and on page 183 they made this claim: I think it must be "when its determinant is $0$" instead of "when its determinant is different from $0$". ...
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36 views

Is it possible to know if a linear transformation is injective,surjective or bijective if it's not a finite dimension?

With finite dimension you can do a matrix representation a of a linear transformation but for example for the linear transformation of the integral operator is it one to one, onto or both? How do you ...
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102 views

Why does $a\cdot r^{-1}$ equate to $\frac {a}{r} = 1$?

Why is $a\cdot r^{-1}=1$ equivalent to $\frac {a}{r} = 1$? I am trying to write exponential functions from graphs; two points were given: $(-1,1)$ & $(-2,5)$. I am trying to find an equation ...
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21 views

Orthogonal lines on Mercator projection?

I am currently struggling with the following task: We have two pairs of latitude/longitude which determine a small line segment It is needed to get two pairs of latitude/longitude for a small line ...
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22 views

Minimal polynomial of projection on a plane?

If $g: \mathbb{R}^3 \to \mathbb{R}^3$ is the projection on a plane, what is the minimum polynomial of g? Related to this what is the minimum polynomial of a reflection with respect to a plane?
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Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
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13 views

Question on the definiton of graded rings

the most common definition of a graded ring $R$ is that $R$ has (as an abelian group) a decomposition as $R=\oplus_{i\in I} R_i$ where $R_i$ are abelian groups and the $\oplus$ denotes the direct sum ...
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Nonnegative solutions of linear equations

I am wondering whether the follow proposition holds or not: Let $A$ be an $n\times k$ matrix with real entries where $k<n$. Suppose that there is some $y\geq 0$ and $y\neq 0$ with $yA^{\...
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Showing expression for $ [ \mathbf{a,b,c}] [ \mathbf{u,v,w}] $ where $ [\mathbf{x,y,z}] $ is the triple scalar product of vectors in $\mathbb{R}^3 $

Any hints/solutions to how I can show $$ [ \mathbf{a,b,c}] [ \mathbf{u,v,w}] = \begin{vmatrix} \mathbf{a.u} & \mathbf{a.v} & \mathbf{a.w} \\ \mathbf{b.u} & \mathbf{b.v} &...
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1answer
30 views

Unable to comprehend results of principal component analysis

(Image below) Image shows the principal component analysis of two points {(1,3), (3,1)}. u1 and u2 are drawn which are corresponding eigenvectors after PCA. The points follow a normal distribution N(0,...
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56 views

Isi B.Math Fibonacci problem.

Let $\dbinom{n}{k}$ denote the binomial coefficient $\frac{n!}{k!(n-k)!}$ , and $F_m$ be the $m^{th}$ Fibonacci number given by $F_1=F_2=1$ and $F_{m+2}=F_m+F_{m+1}$ for all $m\geq 1$. Show that $\...
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1answer
34 views

Maximization problem setup and analysis

So, I needed help in setting up the following: A coffee company sells two types of breakfast blends. They have on hand $132$ kg of dark roast and $84$ kg of hazelnut. One breakfast blend will ...
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Cholesky decomposition of $I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

I need to compute the Cholesky decomposition of the following matrix: $\varPi=I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$ Here $n$ is the dimension of the matrix and $x>0$. $\iota_{n}$ is ...
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What's the name for the following transformation?

Assume $C\in \mathbb{R}^{n\times 1}$. $x\in\mathbb{R}^{n}$, $\Delta\in\mathbb{R}^{n-1}$. There exists a $\Pi\in\mathbb{R}^{n\times (n-1)}$ that: $$x=\Pi\Delta$$ and $$\Delta=\Pi^T x\ and \ Cx=0$$ is ...
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1answer
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What's the name of “the mininum value of $x^TPx$ from the points on a hyperplane to the origin”?

Suppose $P\in \mathbb{R}^{n\times n}$ is a positive definite symmetrical matrix. $F\in \mathbb{R}^{m\times n}$, $g\in \mathbb{R}^{m}$, $n > m$. The set $S=\{x|Fx=g\}$ is an $n-m$ dimensional ...
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69 views

Does this linear system have a single solution

Given unknown $x_1>0$, $x_2>0$, $x_3>0$, $x_4>0$, and known $y_1>0$, $y_2>0$, $y_3>0$, $y_4>0$, $$ \begin{cases} x_1+x_2=y_1 \\ x_1+x_4=y_2 \\ x_3+x_2=y_3 \\ x_3+x_4=y_4 \end{...
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How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
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I need help for steps in how to solve for $L$ [on hold]

$$-(1-L)^{-\frac{1}{2}}L^{\frac{1}{2}} + (1-L)^{\frac{1}{2}}L^{-\frac{1}{2}}=0$$ Thanks in advance, I've been stuck on this for a while. Chris
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1answer
25 views

What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
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Help solving this recurrence relation

I wanted to resolve the determinant of the next (nxn) matrix via recurrence relations: $$ \begin{vmatrix} a & 1 & 0 & 0 & 0 & 0 &.... 0 & 0 & 0 & 0 & 0\\ 1 &...
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Kernel, range, ranks and nullities

A linear mapping is given by the matrix $$\mathit{L}:\mathbb{R}^3\rightarrow\mathbb{R}^3$$ $$\begin{bmatrix} 0 &-1 &1 \\ \frac{1}{2}& -2 &\frac{3}{2} \\ 1& -3 & 2 \end{...
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1answer
18 views

Distributive property on trace norm

I hope this is not a trivial question, basically, if we have trace norm of $A$ defined as $||A||_\star := \operatorname{trace}\left(\sqrt{A^*A}\right) = \sum\limits_{i=1}^{\min\{m,n\}} \sigma_i$, if $...
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Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by the vectors $\begin{bmatrix}2\\2\\-1\end{bmatrix}, \begin{bmatrix}-8\\-2\\5\end{bmatrix}, ...
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1answer
29 views

Determining a basis for Col($A$) and a dimension for the null space of $A$

Let $A = \begin{bmatrix}1&-1&1&0&-2&1\\1&-1&1&1&0&0\\-1&1&-1&2&5&-1\end{bmatrix}$ a) Determine a basis for Col($A$) b) What is the ...
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1answer
41 views

Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
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1answer
34 views

What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix} $$ So here the 1st pivot is missing ...
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1answer
29 views

Find a basis of nullspace(A)

Let $A = \begin{bmatrix}4&-4&2&-6\\2&-2&1&-3\end{bmatrix}$ Find a basis of nullspace$(A)$ I first put $A$ in RREF to get: $\begin{bmatrix}1&-1&1/2&-3/2\\0&0&...
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15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...