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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Finding the Orthogonal Complement to a subspace

So suppose I have a vector space, $V$ which is all continuous functions on $[0,1]$. Additionally, we have an inner product over $V$ where $\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$. Now suppose I ...
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20 views

Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
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1answer
32 views

Help solving the equation [on hold]

I'm stuck and don't know what to do next to solve this equation. Any hints? $y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1)$
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21 views

Which one is equation of tangent

Is equation of tangent plane $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $ or $z=f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $ In my book I found ...
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48 views

Properties of transfer matrices and their traces

I'm having difficulties understanding some arguments in my statistical mechanics lecture and would like to make them more rigorous by proving some properties. For the Ising model on a lattice we ...
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302 views

Algebraic multiplicity = geometric multiplicity?

I was wondering if algebraic multiplicity was equal to the geometric multiplicity. If the matrix (of size $n\times n$) is diagonalisable, i.e. the characteristic polynomial is of the form $$p(x)=(x-\...
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1answer
14 views

Help with some calculations

My question is: what I need to do to get 2nd equation from the first? 1) $TP1 = vp1 · λ + TS1$ $TP2 = vp2 · λ + TS2$ 2)$$TP_2 − TS_2 =\frac{vp2}{vp1}(TP1 − TS1)$$
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1answer
47 views

How to solve proportions involving vector cross products?

I have the following proportion $\vec{JV} \times \vec{F_v} = \vec{JM} \times \vec{F_m}$ and all members are known except the magnitude of the vector $\vec F_m$, like described by another question here ...
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2answers
3k views

$f(a+b)=f(a)+f(b)$ but $f$ is not linear

Can you show me a continuous function $ f \colon \mathbb{R}^n\to\mathbb{R}^m\\ $ that satisfies $f(a+b)=f(a)+f(b)$ but is not linear? We have that $$f(0)=f(0+0)=2f(0)\implies f(0)=0\\ f(x-x)=f(0)=f(...
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46 views

Determine if the following vectors are coplanar.

I have no idea to start with this question, I know how to find if vectors are coplanar when the values of the vectors are given to me, but I do not know how to manipulate coplanarity properties well ...
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17 views

Solving for x in a matrix equation?

I'm confused, how exactly can I solve this? I have no clue where to start Solve for $X$ $\begin{bmatrix}6&8&-6\\1&7&2\end{bmatrix} = 2X - 3\begin{bmatrix}-5&-2&-6\\4&...
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19 views

Find an orthogonal basis for the bilinear form

Find an orthogonal basis for the bilinear form over $\mathbb{R}$ given by $(x,y)\to x^tAy$ where $$A=\begin{pmatrix}1&4&4\\4&4&10\\4&10&16 \end{pmatrix}.$$
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1k views

Change of basis matrix for orthogonal bases

I am trying to show that if $B_1$ and $B_2$ are orthonormal bases for $\mathbb{R}^n$, then the change of basis matrix $P$ from $B_1$ to $B_2$ is an orthogonal matrix. I'm a bit stuck. I started with ...
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35 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
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1answer
31 views

Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
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33 views

Algebraic or geometric multiplicity?

I am reading a proof of the fact that every linear transformation $L:V\to V$ can be represented by an upper triangular matrix $M$, with eigenvalues on the diagonal. And if the algebraic ...
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1answer
31 views

Why this constant $c$ is the determinant of the operator

I'm reading Linear Algebra written by Kenneth M Hoffman and Ray Kunze and on page 172 he states the following corollary: Afterwards, he said that: Of course, the element $c$ in the last ...
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27 views

How long would it take in years to spend 50,000 and only spending 50 dollars a day? [on hold]

A person has won 50,000 dollars and doesn't want to spend it in a short amount of time. Instead this individual has decided to spend 50 each day from the 50,000 he or she has. How long would it take (...
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1answer
43 views

Proof of Isometry and Reflection

$ V = \mathbb R^n$ is provided by the standard scalar product and by the standard basis $S$. $ W \subseteq V $ is a vector subspace and $ W^\bot$ is its orthogonal complement. a) Prove that there ...
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29 views

Number of $4\times3$ matrices of rank 3 over a field with 3 elements.

I am finding number of $4\times3$ matrices of rank 3 over a field with 3 elements. If i count it as number of linearly independent columns i.e $3$ then its answer is $(3^{4}-1)(3^{4}-3)(3^{4}-3^{2}).$ ...
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480 views

Trying to Check Cov Matrix calculation from SVD using Numpy

I am trying to work with the SVD and PCA. Just to check that I am doing what I think I am doing, I did a simple test in in python. The test is that I make a random matrix of realizations, and I ...
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75 views

reflecting a function about a line

Say you have a function $f(x)$ and a line $g(x)=ax+b$. How do you reflect $f$ about $g$? I am apparently supposed to write more text, but the line above is all I am after, hence I wrote this sentence ...
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33 views

Why solving linear equations is taking a quotient by some subspace?

Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations. But this lecture notes suggest ...
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42 views

Finding the equation of a median through a triangle. [on hold]

The line $4x-5y+20=0$ cuts the $y$ axis at $A(0,4)$ and the $x$ axis at $B(-5,0)$. Find the equation of the median through O of triangle $OAB$. Note, the book indicates the answer as $4x-5y=0$ while ...
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33 views

Reducing the Matrix to Reduced Row Echelon Form

Reduce the matrix $\begin{bmatrix}1&-1&-6\\4&-1&-15\\-2&2&12\end{bmatrix}$ to reduced row-echelon form How is my answer incorrect? I performed the row operations: 1) $R_2 =...
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96 views

Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
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15 views

Area of the region bounded by four vectors.

I'm stuck on how to approach this problem. I have a feeling it involves determinants and linear algebra. It's to find the area of the region bounded by the vectors: [-7,7], [5,5], [3, -4], [-5,-6]
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33 views

Linear equation [duplicate]

I have been troubles with the problem below, The line 4x-5y+20=0 cuts the x axis at A(0,4) and the y axis at B(-5,0). Find the equation of the median through O of triangle OAB. Find the equation of ...
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13answers
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Why do we use the word “scalar” and not “number” in Linear Algebra?

During a year and half of studying Linear Algebra in academy, I have never questioned why we use the word "scalar" and not "number". When I started the course our professor said we would use "scalar" ...
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61 views

Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
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36 views

the rational canonical

let T$\in$ $\mathcal{L}$($\mathbb{Q^3}$,$\mathbb{Q^3}$) be given by $$T(v)= \left[ \matrix { 1&-1&-4 \\ 1&-1&-3 \\ -1&2&-2 } \right]v $$ . Find the rational canonical form ...
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25 views

How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
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23 views

Module over nonassociative algebra [on hold]

I have $A$ an algebra $*$ that is commutative an have an identity $(a^2b)b=((ab)ba$ and asking me verify conditions conditions for a $A$-module $*$ $M$. How I would use Linearization of identity? $...
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37 views

Solving the system using elimination

Solve the system using elimination $-6x -2y + 3z = 34$ $-5x -4y + 4z = 32$ $2x +5y -4z = -19$ $x = ?, y = ? , z = ?$ So I threw this in a augmented matrix and put it in REF form ...
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2answers
7k views

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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1answer
25 views

Having trouble understand Row Echelon Form

I'm having trouble understanding Row Echelon Form. I'm trying to solve the system $-2x - 10y - 29z = 5$ $-4x - 19y -56z = -3$ $x + 5y + 15z = 3$ it has the solution $x = ? , y = ? , ...
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1answer
59 views

Calculating inverse function with 2 variables

$f: R^{2}\mapsto R^{2}$ $(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$ I should calculate inverse function of $f$ in point $(3,1)$. I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x ...
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67 views

Points on which function is invertible?

$f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $f(x,y)\mapsto((x-y)^{2}+1, x-y^{3}-2)$ For which points is this function invertible? I calculated the Jacobian matrix, but what should I do next to get ...
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1answer
20 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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1answer
26 views

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $...
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68 views

Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
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50 views

Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues

Let $L:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be defined by $L(A) = A^T,$ where $A^T$ is the transpose of $A$ and $M_n(\mathbb{C})$ is the space of all $n \times n$ matrices with complex entries. Prove ...
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47 views

Matrix with all entries N

Is there a specific name for a matrix where all entries are the name number? I am writing a program where I want to be able to describe a matrix like this in the same way I would the identity matrix, ...
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69 views

How to determine which of the following matrices are similar?

If we have the following three matrices: $$ A=\begin{bmatrix} 7 &1 \\ -5 &3 \end{bmatrix},\;\; B=\begin{bmatrix} 5 &-1 \\ 1 &5 \end{bmatrix},\;\; C=\begin{bmatrix} 5 &1 \\ 1 &...
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65 views

Prove that N(T)=0 and R(S)=U

Let $T:U \to V$; $S:V \to U$ and $ST:U \to U$. Prove that $N(T)=\{0\}$ and $R(S)=U$. My professor gave us a fact at some point that if $ST=ID(U)$ we have S is surjective and T is injective. I am not ...
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57 views

Binary matrices with rank $n$

I'm stuck doing this problem Let $A$ be a matrix of order $n \times n$ with entries in $\{0,1\}$, which has exactly two $1$'s on each row and on each column. Which conditions are necessary and ...
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95 views

For which $a$ and $b$ is this matrix diagonalizable?

For which $a$ and $b$ is this matrix diagonalizable? $$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$ How to get those $a$ and $b$? I calculated ...
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3answers
121 views

Importance of the homogeneity assumption in definition of linear map

Let $V$ and $W$ be vector spaces over field $F$. A function $f: V \rightarrow W$ is said to be linear if for any two vectors $x$ and $y$ in $V$ and any scalar $\alpha\in F$, the following two ...
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87 views

$(AB-BA)^m=I_n$ has solution if and only if $n=mk$ where $m\geq 2$ is an integer number. Is it correct?

I found out that the equation $(AB-BA)^m=I_n$ does have solution when $n=km$, where $k$ is an arbitrary integer number. To prove, we just need to consider $C=$diag($r_1,..., r_m$) where $r_1,..., ...