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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For a linear transformation how can a set be characterized?

Let T be a linear transformation from V to W. Consider the set of vectors S in V s.t. Tx = w where w is a non-zero vector in W. Characterize the set S.
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3answers
25 views

change of basis and inner product in non orthogonal basis

I have some vector, originally expressed in the standard coordinates system, and want to perform a change of basis and find coordinates in another basis, this basis being non-orthogonal. It's ...
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1answer
29 views

Endpoints of a 3D line

How to find the coordinate of the endpoints (A and B) of a line on a surface with known surface normal, center coordinate, and length?
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2answers
35 views

Help me to solve this question. [on hold]

For a particular model of motorcycle the stopping distance at 50 miles per hour is measured at 144 feet and the stopping distance at 70 miles per hour is 296 feet. Using linear interpolation, the ...
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2answers
33 views

What is the general method for solving $Ax = b$ when A is rectangular

Suppose I am given that $A = \begin{bmatrix} 2 &4& 6 &-2 \\ 1& 0& 1 &1 \\ 0& 2 &2& -2 \end{bmatrix}$ and $b = \begin{bmatrix} 4 \\ 2 \\ 1 \end{bmatrix}$ What ...
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0answers
27 views

Help explain “3d algebra”

The following is part of my lecture note, but I get lost after the first paragraph. I know what is "even" permutation and "odd permutation" which I learned from my abstract algebra course, and figured ...
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2answers
39 views

Given two set of eigenvectors, can one be represented by another?

Let $E_1$ be a set of eigenvectors with eigenvalue $\lambda_1$ and $E_2$ be a set of eigenvectors with eigenvalue $\lambda_2$, $\lambda_2 \neq \lambda_1$. Prove or disprove the following statement: ...
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1answer
32 views

The determinant of sum of squares of a special family of real $2\times2$ matrices

Let $A_0, A_1, \ldots, A_n \in M_2(\mathbb R)$ (with $n\ge 2$) be nonzero matrices with the following properties: $$ \begin{cases} A_0 \neq aI_2&\forall a\in\mathbb R,\\ A_0A_k=A_kA_0&\forall ...
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1answer
30 views

Calculate correct dose for pet

if I have a solution that contains 750 mg of curcumin in 10 ml of water and I want to give a dose that is equal to 25 mg curcumin How many ml do I give? Need to medicate my cat and I need to be sure ...
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2answers
44 views

Eigenvalues and power of matrices

Knowing that matrix$$A=\begin{pmatrix}-6&7\\-14&15 \end{pmatrix}$$has eigenvalues $1$ and $8$, find an $2\times2$ matrix $B$ such that $B^3=A$. Usually when I see a question, I would know ...
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0answers
12 views

What does quasi convexity have to do with a bordered hessian matrix?

I understand bordered hessian matrix is used to figure out whether or not the quadratic equation is positive definite, negative definite, indefinite or semi positive definite, semi negative ...
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1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
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1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
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2answers
58 views

Linear Algebra: which of the definition of subspace of a vector space is more correct?

In a test I was asked to give a definition to a subspace to a vector space, I wrote: A subset $V$ is a subspace of $X$ if $0 \in V$ and $\forall u,v \in V, > \exists \thinspace V$ s.t. $ u+v = ...
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1answer
38 views

Why is there 15 principal minors in 4 x 4 matrix?

I have trouble understanding principal minors. Can anyone please help me?
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1answer
12 views

Jordan form transition matrix

Let $A$ be a matrix with eigenvalues 1 and 0 only. Suppose $J=PAP^{-1}$ where $J$ is the Jordan form. How to show that $P$ only have entries 0 and 1? Any hints, idea?
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5answers
65 views

Finding amounts of ingredients in a food based on nutrients

Imagine that there are 50 ingredients, $I_{(1-50)}$, cake can possibly be made out of. Our friend makes a cake from unknown amounts of these ingredients. Therefore the cake $C$ is composed as such: ...
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2answers
42 views

A bound on entries, and a bound on the determinant

Let $A$ be a $3\times 3$ real matrix with all $0\leq a_{ij}\leq 1$. Show that $\det A\leq 2$. So the hint my classmates give was to assume all $a_{ij}=\{1,0\}$. Why is that? Why does considering ...
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1answer
31 views

Linear Map Involving a Polynomial [on hold]

How do you prove that the linear map p -> (p(-1) p(0) p(1)) is injective? I know you have to show that the kernel = 0 but how the p(-1) etc is confusing me.
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1answer
21 views

Jacobian Eigenvalue Algorithm and Positive definiteness of Eigenvalue matrix

For a real symmetric matrix A of size n x n, the Jacobian Eigenvalue Algorithm produces n - Eigen values of A in the form of a Square Diagonal Eigenvalue Matrix of order n n - Eigen vectors of A ...
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3answers
87 views

Do mathematicians prefer eigenvectors with purely integer entries?

I was solving a trivial linear algebra question. Suppose we have $\begin{bmatrix} 1 & 3 \\ 5 & 3 \end{bmatrix}$, find all eigenvectors. Okay, so one of its eigenvecctor is $\begin{bmatrix} ...
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3answers
55 views

Solving linear system of equations to obtain different classes of solution.

Correct me if I am wrong. Find the value(s) of the constant $k$ such that the system of linear equations $$\left\{\begin{array}{l} x + 2y = 1\\[2ex] k^2x − 2ky = k + 2 \end{array} \right.$$ has: ...
2
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1answer
27 views

Integrality Test: Matrices as Homomorphisms of Modules

I am reading in Neukirch's Algebraic Number Theory book and I find an argument which I stumble upon in several places and which I do not understand. The discussion is fairly at the beginning and the ...
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0answers
35 views

Inverse of $A^\top BA+C^\top DC$?

I'm numerically solving a system of equations of the form: $$Mx = b$$ where: $$M = A^\top BA+C^\top DC,$$ $B$ and $D$ are block-diagonal, $A$ and $C$ are $n\times m$ matrices with $m \leq n+3$. ...
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1answer
22 views

Minimal polynomial with repeated factors over an algebraically closed field.

Let $k$ be an algebraically closed field and let $V$ be a vector space over $k$ and let $T: V \to V$ be any linear transformation. I can't think of an example when the minimal polynomial of $T$ will ...
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2answers
42 views

Vector space sizes

$T:V \rightarrow V$ where $V$ is a finite dimensional real vector space I can show that $\ker(T) \subseteq \ker(T^2) \subseteq \ker(T^3) \subseteq\cdots $ Prove there exists some $k$ such that ...
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2answers
43 views

Number of Nilpotent matrices.

I am trying to find number of nilpotent matrices over the field of reals with entries as $0$ and $1$ only. I tried it for order $2$ its comes only $2$ as diagonal elements are always $0$. Is there ...
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0answers
34 views

Number of Idempotent matrices.

I am trying to find number of of Idempotent matrices of order $n$ with entries as only $0$ and $1$ over the field of reals. I tried it for matrices of order 2 its comes 8. Is there any general method ...
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1answer
26 views

A complex matrix with real eigenvalues

Let $A$ be a $10\times 10$ matrix with complex entries and all eigenvalues non-negative real numbers and at least one eigenvalue strictly positive . Then there exist a matrix $B$ ...
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1answer
20 views

Are scalar/vector fields in multivariable calculus related to fields of vector spaces in linear algebra

In linear algebra, I have learned that vector spaces are defined over fields. I have to admit that I don't have any background in abstract algebra, so my knowledge of fields are limited to $\mathbb R, ...
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1answer
43 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
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2answers
34 views

$A$ is an invertible matrix such that the sum of each row is $1$ [duplicate]

Let $A$ be an $10\times 10$ invertible matrix with real entries such that the sum of each row is $1$. Then $A$. The sum of entries of each row of the inverse of $A$ is $1$ ...
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0answers
24 views

Prove that this matrix is total unimodular

Is there an easy way to prove that this matrix is total unimodular ? $$ \begin{bmatrix} 1 & F_1 & 0\\ 1 & 0 & F^T_1 \\ 0 & F_2 \end{bmatrix} $$ $1$ is the identity matrix, ...
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3answers
39 views

Is $A(x, y, z) = (xy + z, yz - x)$ a linear function?

Is this a Linear function? If yes, what is its matrix? $$A:\mathbb{R}^3\to\mathbb{R}^2, A(x, y, z) = (xy + z, yz - x);$$ Since they are all first degree variables, i think that this is a linear ...
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2answers
27 views

Rigorous Linear Transformation Proof

$T:V \rightarrow V$ We could also write: $T:V \rightarrow Im(T)$ The question tells us that $Im(T)=Im(T^2)$ It's intuitively obvious that this means that T then maps $Im(T)$ to itself so if you ...
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1answer
21 views

orthogonality - which vector in the subspace W is closest with y

I'm having some difficulties by calculating the vector which is closest with y. We have 2 vectors and y and the question is: which vector in W = span(u1, u2) is closest with y. where: $$u1 = ...
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0answers
34 views

Generalized inverse of matrix product involving a positive semi-definite matrix

I have the following: A real square positive definite matrix $A$, and a real square conformable positive semi-definite matrix $B$. I form the product $$C = A^{-1}BA^{-1}$$ and I wonder, is it true ...
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1answer
41 views

Solving linear system of equations to obtain different classes of solution [closed]

Find the value(s) of the constant $k$ such that the system of linear equations $$\left\{\begin{array}{l} x + 2y = 1\\[2ex] k^2x − 2ky = k + 2 \end{array} \right.$$ has: No solution An infinite ...
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2answers
27 views

Simple Ratio Question - What is wrong with my approach?

A zoo has twice as many zebras as lions and four times as many monkeys as zebras Their total is a multiple of ? (The answer is 11 but my solution gets me 13) Here is how I'm solving it: Z=2(L) -> ...
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1answer
39 views

Finding eigenvalues and eigenvectors

Let $v=(3 ,1 ,3 ,-4)$ and $A=v^t\cdot v$. Find the eigenvalues and eigenvectors of $A$ without calculating the whole matrix $A$ $Rank(AB) \leq min((Rank(A),Rank(B))$ so $Rank(A)\leq 1$ but ...
3
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1answer
47 views

graph partition, second smallest eigenvalue.

In spectral graph partition theory, the eigenvector corresponding to the second smallest eigenvalue of the laplacian matrix of a graph, in general, is used to partition the graph. What is the ...
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3answers
172 views

Solving geometry problem, in a triangle, using vectors

P is the middle of the median line from vertex A, of ABC triangle. Q is the point of intersection between lines AC and BP.
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2answers
47 views

Finding the characteristic polynomial

let $A$ be \begin{pmatrix} 7 & -5 & -4 \\ -1 & -2 & 1\\ 9 & -5 & -6 \end{pmatrix} find the characteristic polynomial, eigenvalues and eigenvectors So we ...
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1answer
38 views

Number of involutory matrices of order n.

I am trying to find number of involutory matrices $(A^{2}=I)$ of order $n$ with entries as only $0$ and $1$ over the field of reals. But i did not get any formula of it. For order $2$ there are only ...
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0answers
37 views

If $\mathcal{B}_1$ and $\mathcal{B}_2$ are two bases for $V$ then $\#\mathcal{B}_1 = \#\mathcal{B}_2$ [closed]

If $\mathcal{B}_1$ and $\mathcal{B}_2$ are two bases for $V$ then $\#\mathcal{B}_1 = \#\mathcal{B}_2$ I get the intuition, but how do you prove it with mathematical notations?
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0answers
31 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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0answers
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$A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\} \Rightarrow$A is closed [closed]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
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2answers
332 views

A matrix with the highest number of eigenvalues

Is is true to say that the matrix with the highest number of eigenvalues is the scalar matrix?
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2answers
33 views

systems of equations with 3 variables - addion method

I am stuck on solving the following systems of equations with 3 variables. The textbook asks to use the addition method so can we please stick to that. ${5x -y = 3}$ ${3x + z = 11}$ ${y - 2z = ...
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2answers
51 views

one to one and onto meaning

If $T$ is a linear transformation and is said to be one to one or onto- this only makes sense when we specify what domain and range is right? $T: V \rightarrow V$ may not be onto or one to one but ...