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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Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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14 views

Nonnegative solution to underdetermined linear system

I would like to show that the underdetermined system $Ax=b,\; x\ge 0$, with $b$ being a positive vector and $A$ being a binary matrix, has at least one solution. I've seen several other related ...
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1answer
17 views

Solving a system of xor equations?

How can I solve the following system of xor equations? k0 ⊕ k2 ⊕ k3 = 0011 k0 ⊕ k2 ⊕ k4 = 1010 k0 ⊕ k1 ⊕ k2 ⊕ k3 = 0110 How can I solve this system to know the ...
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1answer
51 views

what does $p(-1) = 0$ mean?

In a linear algebra problem, it asks me to determine the subespace spanned by $$ \left\{ p(x) \in \mathbb{R}^3 : p(-1) = 0 \right\}. $$ What does it mean?
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23 views

In $\mathbb{R}^{3}$, does an orthogonal basis of integer vectors exist such that none of their coordinates is $0$?

In $\mathbb{R}^{3}$, does an orthogonal basis $\{$ $(a_{1}, a_{2}, a_{3}),$ $(b_{1}, b_{2}, b_{3}),$ $(c_{1}, c_{2}, c_{3})$ $\}$ exist such that all $a_{i}, b_{i}, c_{i}$ are integers $\neq 0$?
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25 views

Characteristic Polynomial Calculation

I have a problem in my homework in which I have to find the characteristic polynomial of the following matrix: I know the final solution is: However, my answer keeps getting wrong whenever I ...
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1answer
36 views

Finding linear transformations. [on hold]

Can somebody provide some idea on how to tackle this problem? Find all linear transformations $$T:\mathbb{R}^3\to\mathbb{R}^2,\ (u,v,w)\to(x,y)$$ which map the $u-w$ plane onto the line given by the ...
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1answer
17 views

Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
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39 views

Jordan normal form book

I am currently reading the book Basic Algebra [modern] Anthony W. Knapp about Jordan canonical form Is there any detailed oriented book about Jordan Normal Form which explain : An Algorithm to put ...
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46 views

I need help in Algebra. [on hold]

can someone explain to me the Vector space and the Vector base? (their uses and how to find a matrix' base and space and so on..) thanks in advance
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27 views

Jordan normal form Upper or lower

I am reading a jordan form book at the moment, Basic Algebra [modern] Anthony W. Knapp page $231$, but I feel the lack of understanding : should we have to start with the Bigger Jordan blocks of ...
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2answers
95 views

Probability of building an Invertible Matrix

If we build a 10X10 matrix,randomly filling with 1's and 0's, is it more likely to be invertible or singular? First of all until we have 10 1's its not going to be invertible. With 10 1's on the ...
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27 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
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1answer
20 views

Vectors components that are not contra or covariant?

I know that a vector can have contravariant and covariant components, but is it possible to have components that are neither contravarient or covariant? I suspect that the answer is yes, and that most ...
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4answers
72 views

If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$ .

Having trouble solving this. If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$. .
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5 views

How to find the long range transition matrix L of P

P is the transition matrix of a regular Markov chain. Find the long range transition matrix L of P. $$ P = \begin{bmatrix} 1/2 & 1/4 & 1/4\\1/2&1/2 &1/4\\0 &1/4 & ...
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3answers
42 views

Property of SO(3)

Suppose $A\in SO(3).$ Show that there exists a vector $v\in \mathbb{R}^3$ such that $Av=v$. $ SO(3)={{A\in O(3)|detA=1}} $ and $ O(3)={A:\mathbb{R}^3\rightarrow ...
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1answer
24 views

Prove that the columns of the first matrix span but the columns of the second matrix do not span.

A = [1 0 1 0] = row 1 [1 2 0 1] = row 2 (2 * 4 matrix) and [0 0] = row 1 [2 1] = row 2 (2 * 2 matrix) I know that Column of matrix of m*n dimension spans if rank of matrix is equal to m. ...
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2answers
39 views

What are the standard defintions of “counterclockwise” and “clockwise” in 3d space?

I'm in Calc III right now, and I'm a little confused as to what constitutes "clockwise", and "counterclockwise" rotations when dealing with the various planes in 3d-space. Of course, it's obvious in ...
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1answer
33 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
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1answer
20 views

nullity and rank of the linear transformation $T: T [ p (x)]= p(x+1)$

Let $V$ be the linear space of all polynomials $p(x)$ of degree $\le n$. if $p$ belongs to $V$ and $q = T(p)$, means that $q(x) = p(x+1)$ for all real $x$. find nullity and rank of the linear ...
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2answers
23 views

Find the matrix $M$, given four vectors

If you have $4$ vectors in a plane $x_1, x_2, b_1, b_2$, and a matrix $M$ such that $Mx_1 = b_1$ and $Mx_2 = b_2$, how do you find $M$ from this given data? Any hints would be appreciated; I am not ...
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1answer
29 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
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1answer
29 views

Nullity and rank of the linear transformation $T[f(t)] = \int_a^b f(t) \sin (x-t) dt ~\forall~x \in [a,b]$

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ Then, the nullity and rank ...
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0answers
26 views

Algebraic multiplicity of an eigen value

Let $T$ be an operator on a complex Vector space $V$. Then, the algebraic multiplicity of an eigen value is equal to $\dim ~null~ (T - \lambda I)^{\dim V}$ Which means, if we obtain the upper ...
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2answers
79 views

$x=x^2$ in a sub group?

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
2
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1answer
21 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
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26 views

Linear maps and subspaces

The set-up for my question is this, let $k \le n$, let $E \subseteq \mathbf{R}^n$ be a $k$-dimensional subspace. Let $I \subseteq \{1,\ldots, n\}$ such that $|I| = k$, then we can define coordinate ...
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35 views

How to calculate time-of-flight and target hit point of a ball thrown against a wall?

Imagine you are throwing a ball against a distant wall, the question is how to find the time taken by the ball to reach the wall and also the point of impact on the wall (after the ball has bounced ...
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1answer
18 views

Adjoint Transformations and Self-Adjoint Operators

I don't quite understand the whole adjoint and self adjoint thing. I know their definitions: Given a linear transformation $A:\mathbb{R}^d \to \mathbb{R}^m$, its adjoint >transformation, ...
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2answers
27 views

Characteristic polynomial of a linear endomorphism of dimension $n$.

So, if $T: V \rightarrow V$ and I suppose that $T^2-3T+2I=0$ and that the $\mathrm{rank}(I-T)=k$, what would be the characteristic polynomial of $T$? I know from previous questions that the ...
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1answer
17 views

Inversing badly-conditioned square matrix: methodology

I have a badly-conditioned square matrix. I need to inverse it. For inversing, currently I'm doing the following steps: I take the badly-conditioned matrix with size of $n$ by $n$ By reduced row ...
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1answer
19 views

Derivative over scalar field with respect to fixed point proof.

Prove there is no such scalar field that $f '(a;y) >0$ for fixed point $a$ and every non-zero vector $y$. I posted this question but some of you pointed out that it is not clear. So, $f ' (a;y)$ ...
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1answer
11 views

Projective transformation

I need to find the function $f$ that satisfies the following: $f((1:1:0))=(0:1:1)$ $f((0:1:1))=(1:0:1)$ $f((1:0:1))=(1:1:0)$ If I let: $x=(1:1:0)$ $y=(0:1:1)$ $z=(1:0:1)$, then I get $f(x)=y$, ...
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2answers
32 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
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1answer
38 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
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16 views

See if vector set is basis of space using Gram Schmidt process

I have a problem my teacher gave me and I can't find an answer. She gave me a set of 3 vectors, $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 7\\3\\1 \end{bmatrix} \begin{bmatrix} ...
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1answer
21 views

Prove that the LDU factoriztion is unique [on hold]

How would one prove that the LDU factorization of a matrix is unique?
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2answers
20 views

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that a basis for $V$ need not contain a basis for $S$.

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that every basis of $S$ is part of a basis for $V$ but a basis for $V$ need not contain a basis for $S$. Attempt: ...
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3answers
49 views

Sufficient and necessary conditions to obtain a solution

Find sufficient and necessary conditions for which the following system of equations: $$ax+by=c$$ $$dx+sy=h$$ $$qx+wy=v$$ has at least one real solution $(x,y)$. Here $a,b,c,d,s,h,q,w,v$ are real ...
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True or False. The intersection of any two subset of vector $V$ is a subspace of vector $V$. [on hold]

The intersection of any two subsets of a vector space $V$ is a subspace of the vector space $V$. Explain if it is true or false.
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1answer
18 views

Maximizing the number of zero entries in a linear combination of matrices

I was wondering if there exists an algorithmic way of solving the following problem. Let's say you have a bunch of square $N\times N$ matrices (call them $M_i$), and you want to form a linear ...
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34 views

Please guide me what are the topics i need to study in maths from basic. [on hold]

I am not having good knowledge in maths.Please guide me what are the topics i.e (algebra,calculus,diff.eqn...)i need to study by step by step. please guide me.
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2answers
57 views

Find a positive definite matrix B such that $B^2=A$. [on hold]

Find a positive definite matrix B such that $B^2=A$, where $$A=\begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}$$
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1answer
22 views

How to prove that $B$ is positive definite when $\|A-B\|\leq\lambda_\min(A)$ for some positive definite $A$?

Denote by $\mathbb R^{n \times n}$ the vector space of $n \times n$ matrices with real entries. For $A \in \mathbb R^{n \times n}$, the notation $A\succ 0$ means that $A$ is symmetric and positive ...
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1answer
20 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
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2answers
263 views

Why does Gaussian elimination not preserve similarity of a matrix?

I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3. In it, the author states that one does not use Gaussian elimination for ...
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2answers
32 views

Linear transformation Df=$\frac{df}{dx}$

Let $Rx$ define vector space of all real polynomials. Let $D:Rx \to Rx$ denote map Df=$\frac{df}{dx}$, for every f. Then which of following is true. $D$ is one-to-one $D$ is onto There exist $E:Rx ...
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0answers
19 views

Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers)

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
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0answers
27 views

Proof the $\mathrm{rank(rows)=rank(columns)}$

Assume we have matrices $A=BC$.It is obvious that the $i$th row of $A$ is a linear combination of the rows of $C$ with coefficients from the $i$th row of $B$ or $b_{i1}C_1........b_{in}C_n$. ...