Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Quadratic form in canonical form relation

The homogeneous quadratic equation can be written as a matrix. It is also written as a canonical form by using orthogonal transformation. Why we are going for canonical form and what is the relation ...
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29 views

Calculating Vandermonde determinant

I understand that the Vandermonde determinant $$ W(x_1, \ldots, x_n) = \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & ...
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1answer
20 views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
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1answer
43 views

When and why can functions “take on” the role of vectors in defining vector speaces?

In what I call "advanced" linear algebra, we examine the properties of vectors in a vector space like an inner product space by checking that they satisfy e.g. the Cauchy-Schwartz inequality, the ...
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1answer
21 views

Describe the solution set of the system

Consider the linear system below: $$\begin{array}{ccccccc} x_1&-&2x_2&+&&&x_4&=&1\\ 2x_1& -& 5x_2& -& 2x_3& +& k^2x_4 &= &-2\\ ...
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Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
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4answers
52 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
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1answer
28 views

Matrix Multiplication - When do you only multiply by one number and add vs. multiplying all numbers?

*I wasn't sure where to put this. Just let me know if I should delete it or if there is another category/website where this question would fit better. Thanks! Or if you know the answer & don't ...
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0answers
49 views

Inverting the infinite matrix with entries $\mathbf{P}_{ij}={i-1\choose j-1}$

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
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1answer
27 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
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1answer
35 views

Hamming Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
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1answer
54 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
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2answers
26 views

problem with denominator in transformation

hi i cant understand where the 2 comes from in this transformation any help would be appreciateD
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1answer
36 views

calculus / algebra

Hi can anyone go through the transformation of the equation below as i cannot understand where the 2 in comes from any help would be much appreciated $$\frac{\omega k^{0.5}}{\omega k} = ...
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Positive definite [on hold]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
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Polynomial Discriminant of 4*g+h^2

Suppose we have two polynomials $g,h\in \mathbb Z[x]$ with $\deg g = 2k+1 =:n$ and $\deg h=k$. As an example, take $g=x^7+2\cdot x^6+x+2$, $h=x^3+x+7$. My question is: Why does the discriminant of ...
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1answer
22 views

Determinant of a rank-one update of a scalar matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
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Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
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1answer
28 views

Positive definite matrix.

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
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0answers
17 views

Form the biquadratic equation two of whose roots are i and 3 . [on hold]

plese answer this quikly Form the biquadratic equation two of whose roots are i and 3 .
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T has zero as a characteristic root [on hold]

Let $ V $ be a vector space over field $F$. $T$:$V$$\rightarrow$$V$linear transformation such that $T$ has zero as a characteristic root.Then 1.T is diagonalisable over $F$ 2.Multiplicity of each ...
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37 views

Subset of vector space containing zero vector. [on hold]

Subset of vector space containing zero vector is always linearly independent.Is this statement is true?
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2answers
32 views

Proof that Every Positive Operator on V has a Unique Positive Square Root

Suppose V is a finite-dimensional, nonzero, inner-product space over F, and F denotes R or C. My thought is : suppose T is a positive operator; thus, T is self-adjoint. Every self-adjoint operator on ...
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22 views

Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important ...
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2answers
71 views

Prove there is a subspace of $V$ isomorphic to $T(V)$

If $T:V\to V$ is a linear transformation and $T(V)$ is of finite dimension then prove that there is a subspace $U$ of $V$ isomorphic to $T(V)$ and then show that, if $x,y\in V$, then $(x+U)\cap ...
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52 views

Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however: A standard definition of linear function (or linear map) is: $$f(x+y)=f(x)+f(y),$$ ...
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1answer
22 views

Identifying if a $ S $ given is a vector subspace

Could you help me to identify if $ S $ is a vector subspace? I started learning linear algebra and I get this question and I am lost.
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38 views

Show that vectors of the form (a,b,1) do not form a vector space

Show that vectors of the form $(a,b,1)$ do not form a vector space I tried using the vector space axioms to attack the problem but I am not going anywhere with this problem. I do not need a ...
3
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1answer
39 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
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2answers
48 views

If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
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35 views

How many Geese were there before any flew away? [on hold]

This equation represents geese flying away in one hour intervals. How many geese were there before any flew away? The first part x - [1/5 = x] represents the quantity of geese flying away at 1:00pm ...
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1answer
22 views

Inverse Matrix Multiplication

Let $A \in F^{n*n}$ a inverse matrix and $B\in F^{n*n}$ a none inverse matrix We can say that because A is row equivilate to $I_n$$ \implies $ $AB$ is none inverse matrix?
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33 views

Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}} $ equal? ...
3
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1answer
43 views

Special solutions to Ax = 0

I solved most of it, just not sure about one point. The problem statement, all given variables and data Suppose A is the matrix shown below: $$ \begin{pmatrix} 0 & 1 & 2 ...
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1answer
31 views

Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
2
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1answer
59 views

Proof $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular

I need help proving $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular, as part of the proof of the Cayley-Hamilton theorem The result makes a lot of sense but I can't prove it ...
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1answer
42 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
3
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1answer
86 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
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0answers
32 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
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19 views

Rank Factorization

I read the following proof Let A, B be m × n matrices. Then rank(A + B) ≤ rankA + rankB. Proof: Let A = XY,B = UV be rank factorizations of A, B. Then A + B = XY + UV = [X,U][Y V] Therefore, ...
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1answer
24 views

Subspace of a vector space Definition

If $W$ is a subspace of a finite-dimensional vector space $V$, then: $\dim(W) \leq \dim(V)$. That makes me think about the definition of a subspace. For example, in $\Bbb R^3$, is $\Bbb R^3$ ...
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43 views

Matrices over field with characteristic $p$

For $A,B$ $n\times n $ matrices over a field $F$ with characteristic $p$ if $AB-BA=cI$ for $c\in F$ does this imply that $c=0$? Intuitively I would say that it doesn't but I cannot think of a ...
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2answers
50 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
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1answer
26 views

Find the kernel of linear transformation

Linear transformation $l:\mathbb{R}^3 \mapsto \mathbb{R}$ is determined as follows: $l(1,0,0)=1$; $l(1,4,0)=-1$; $l(0,0,1)=1$. I need to find $\text{Ker}(l)$. Answer should be ...
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1answer
39 views

What is the space spanned by $a\cos x + b\sin x$?

Consider the case when $x = \pi/4$. $\cos \pi/4 = 1 = \sin \pi/4$. Now, if $a = 1$ and $b = -1$, $a\cos x + b\sin x = 0 $(for non-zero a and b). Does this imply that $\cos x$ and $\sin x$ ARE ...
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30 views

Circular Matrix Linear Independence

Suppose I have the following $N\times N$ circular matrix: $$ \left[ \begin{matrix} 0 & 1 & 2 & .... & N \\ N & 0 & 1 & .... & N-1 \\ . & . ...
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1answer
18 views

Lie algebra for SO(3) as a skew symmetric matrix

How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
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Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
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2answers
67 views

Visually understanding the formula for the distance from a point to plane.

Ok, so we know that if we have an arbitrary point, $p$, and a normal perpendicular to an arbitrary plane, $n$, the distance from the point to the plane can be computed as follows: $$distance = p ...
3
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2answers
46 views

What is the determinant value of $J-I$ if $I$ is identity matrix and $J=(1)_{101\times 101}$? [duplicate]

Let $J$ be a matrix of order $101\times 101$ which each entry is 1 and suppose $I_{101}$ is identity matrix of order $101\times 101$. The question is : what should be the determinant value of $J-I$ ? ...