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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$Ax = b$ & $Ax + b$

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
11 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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2answers
32 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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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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32 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
13 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
35 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
11 views

equation system: solve $L_{1,1}$

a and b is two given constants, let L_(a,b) denote the system x1 2x2 3x3 = 0 2x1 4x2 ax3 = 0 3x1 bx2 9x3 = ab i) solve $L_{1,1}$ ii) find the pairs ...
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1answer
31 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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18 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
27 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
27 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
25 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
72 views

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

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
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1answer
19 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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23 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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1answer
30 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
19 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
15 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
17 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
10 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
28 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
33 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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0answers
15 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
18 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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19 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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36 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
17 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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32 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
53 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
17 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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2answers
261 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
25 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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16 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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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 $b_{i1}C_1........b_{in}C_n$. ...
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1answer
19 views

Injective acton of an Chevalley generator of $\mathfrak{sl}(2)$ on non-intergral weight module

I have a problem as follows. Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. ...
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1answer
41 views

Proof of dimension equality

Is my proof of $\mathrm{dim}(U+W)=\mathrm{dim}(U)+\mathrm{dim}(W)-\mathrm{dim}(U \cap W )$ correct?.Suppose we the basis of $U \cap W$ is $B_{0}$ then we can add this basis to the basis of W let ...
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How to prove that the limit of solution is $-\pi^2/3$?

Consider a matrix $M$ with $2N+1$ rows and columns, so that for $p=0,1,...,2N$ and $k=-N,...,N$ matrix elements ($k$ indexes columns, $p$ is index of row) are $$M_{pk}=\frac{k^p}{p!}.$$ Taking a ...
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1answer
23 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
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1answer
25 views

Finding linear independence in $v_1,\ldots,v_m$

First, I'll try not to ramble, although it tends to happen when I type. I have the following linear algebra problem for my homework. Prove or give a counterexample: If $v_1, v_2, \ldots , v_m$ are ...
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1answer
42 views

Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. [duplicate]

so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant ...
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1answer
16 views

Determining optimal size of rectangle to maximise volume

I am having problems understanding how to solve this question, any help would be much appreciated. An open box (no top) is made from a rectangular sheet of cardboard that measures 20 cm by 30 cm by ...
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2answers
36 views

Determine the kernel of a linear map $f:U \to V$

Let $U=<\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & ...
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12 views

Can anyone help me prove derivative of scalar field using mean value theorem?

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x ...
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2answers
34 views

General form of an element of the othogonal basis of $q$

Let $$q \begin{pmatrix}a & b \\ c & d\end{pmatrix}= (a-b)^2+(b-c)^2+(c-d)^2$$ quadratic form on $M_2(\mathbb{R})$. How can I prove that every orthogonal basis $B$ of $M_2(\mathbb{R})$ has ...
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3answers
37 views

Diagonalizability of endomorphism $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$.

Let $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$. How can I determine what is the explicit expression of $f$, and, most importantly, how do I see if it is diagonalizable? The ...
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2answers
29 views

Alternative proof of a transpose property

I am asked to prove; $$(AB)^T=B^TA^T$$ although it is very simple to prove it by the straight forward way, in the exercise I am asked to prove it without using subscripts and sums, directly from the ...