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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If $\phi(v_1),…\phi(v_\rho)$ are linearly independent, show that $v_1,…,v_\rho$ are linearly independent

Let $\phi:V\rightarrow W$ be linear. Suppose that $v_1,...,v_\rho \in V$ are such that $\phi(v_1),...\phi(v_\rho)$ are linearly independent in $W$. Show that $v_1,...,v_\rho$ are linearly independent. ...
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1answer
19 views

help clearing the doubt about column space

it is an easy one but I don't have anyone else to ask rather than stack exchange doubt is it true that if $b\in \mathbb{R^{n}}$ such that $b\in col(A)$ where $A$ is n×m matrix then there exist unique ...
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1answer
31 views

Generating a basis of the orthocomplement of a vector in $\mathbb{E}^{n}$

There is a vector $v\in\mathbb{E}^{4}$. I can define an orthogonal complement with basis vectors $e_{1},e_{2},e_{3}$ by the rule: $$e_{i}=P_{i}\cdot v,$$ where $$P_{1}=\begin{pmatrix}0 & 0 ...
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0answers
20 views

non negative matrix factorisation [on hold]

i am working on a project involving the use of non negative matrix (NMF) for the separation of a mixture of audio signals.can please give me the mathematical explanation of NMF so that i can ...
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0answers
17 views

What are the different ways of performing Triangular matrix-vector multiplication?

Suppose we have $$\left[\begin{array}{cccc} x_1 & 0 & 0 & 0 \\ x_2 & x_1 & 0 & 0 \\ x_3 & x_2 & x_1 & 0 \\ x_4 & x_3 & x_2 & x_1 \end{array}\right] ...
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0answers
37 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
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35 views

Can these triangle shapes be added up? what is it? is it in 2d or 3d? [on hold]

okay let me ask it a different way, if i make the statement you can never count the amount of triangles you can see in the picture because of the fact that you can see the triangles in both 2D and 3D ...
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2answers
24 views

Cardinality of the set $S$ where $S=\{T:\Bbb{R}^3\to \Bbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\}$

Let $S=\{\,T\colon \mathbb{R}^3\to \mathbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\,\}$. Then $S$ is A. a singleton set B. a finite set containing ...
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1answer
17 views

Finding null space of symmetric matrix generated by outer product

Let $p, q \in \mathbb{R}^n$ such that $||p|| = ||q|| = 1$ and define $A = pq^T + qp^T$. I am trying to find the null space of $A$, but am not having very much luck. I have managed to show that $p + q$ ...
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3answers
54 views

Find the value of $x$ which is correct

I have one exercise which is $$(x+2013)(x+2014)(x+2015)(x+2016)+1=0$$ I tag $A=x+2013$ or other for many ways but still can not find the first $x$ value. please help.
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0answers
55 views

Linear Algebra book for beginners

sorry, if this question is redundant, but I think my case is a bit different. I'm using videos on Khan's Academy to self-study Linear Algebra.And I also bought Linear Algebra And its Applications. By ...
3
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1answer
25 views

Proving that $V = U \oplus W$ where $W$ and $U$ are sets of eigenvectors of $S: V \to V$

Let $V$ be a finite dimensional real vector space, $S : V \to V$ be a linear map such that $S^2 = I$. Show that $V = U \oplus W$ where $U = \{u \in V : Su = u\}$ and $W = \{ w \in V : Sw = -w\}$. ...
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0answers
25 views

Calculate Rotation and Translation Matrix to align elements of input matrix A to Target matrix B in 2d?

I have a matrix in 2D space; the matrix contains elements which I would like to translate into the center of the matrix. Then, I would like to rotate these elements (I mean the positions of the ...
2
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1answer
15 views

Finding char polynomial in $Z_3$

$ K=Z_3 $ $ A \in K_{(4 \times 4)} $ $$A= \begin{bmatrix} a & -1 & -2 & -2 \\ 0 & a-1 & -2 & 0 \\ -2 & 0 & a & 0 \\ -2 & -1 & 0 & a-2 \\ ...
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0answers
12 views

Understanding if the definition of constant normal set depends on the choice of the scalar product or not

Suppose we have a Lie group on $\mathbb R^n$, let's say $(\mathbb R^n,*)$. Suppose also that its Lie algebra $\mathfrak g$ is stratified: I mean that there exists a decomposition of $\mathfrak g$ as ...
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0answers
22 views

Extending the trace inner product to all matrix (real) inner products

In ${\bf R}^{n\times p}$ we have the trace inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. All inner ...
2
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1answer
21 views

Graph the straight line corresponding to the rule (y=7x) for 0≤x≤15

I have attempted this question but I don't really know where to even start. I have graphed y=7x but i'm not sure where to go from there. I am a bit stuck on graphing a line that is relating to 0≤x≤15. ...
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1answer
26 views

Is the Probability of Selecting 3 Random and Colinear Points nil?

Recently, the mathematics YouTube channel released a video titled "Triangles have a Magic Highway - Numberphile". In the video, at 6:40, the expert being videoed says that the probability of any three ...
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2answers
66 views

advanced solutions for an elementary problem!

Let $A\in M_3(\mathbb{R})$ be a matrix of rank one. Suppose that the first row of $A$ is an eigen vector of $A$. I want to show that $A$ is symmetric. My attemp: Actually its simple, for example ...
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1answer
21 views

Sum of component projection matrices

Show that if $X$ $=$ [$X_1$ $X_2]$ and $X_1'X_2 = 0$, then $P = P_1 + P_2$, where $P$ is defined as $X(X'X)^{-1}X'$, the projection matrix. Don't quite know where to start. I tried evaluating it by ...
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Finding The Orthogonal Projection of a Vector Onto a Subspace

Find the orthogonal projection of the vector $\vec{v}=16(1,0,0,0,0,1,1,1,1,1)$ onto the subspace spanned by the vectors $\vec{u1}=16(0,1,0,0,0,1,1,1,1,1)$ and $\vec{u2}=16(0,0,1,0,0,1,1,1,1,1)$. I've ...
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A little help with inequalities (homework) . [on hold]

Can anyone please verify my answers for this problem? $a,b,c,d$ are rational numbers. Answers should be true or false. If $a<b$ and $c<d$ then $\frac{a}{c} < \frac{b}{d}$ to which my answer ...
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0answers
20 views

Relation between the eigenvalues of A and its Hermitian matrix

What is the relation between the eigenvalues of A (given by $\lambda_1$, $\lambda_2$, ...,$\lambda_n$) and the eigenvalues of AH (Hermitian matrix of A)?
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2answers
36 views

Jordan form of the matrix $\left(\begin{smallmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{smallmatrix}\right)$

Determine the Jordan form of the matrix $A= \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}.$ I've calculated the characteristic polynomial ...
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1answer
29 views

Products of Symmetric Matrices

I am trying to find an example of two matrices A and B such that AB is invertible, but A and B are not. I have an example for the singular 1x1 matrix, because I can take $A=(1,0)$ and $B=(1,0)^T$. ...
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1answer
20 views

vector dot product

I have the multiple choice questions (from a past exam, not for marks don't worry) that states: If $u$ and $v$ are vectors such that $\| u+v \| = 2$ and $\| u-v \|= \sqrt{8}$, then the dot product of ...
2
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1answer
32 views

Show that $X_1^T[I_n - X({X^T}X)^{-1}X^T] = 0$

Where $X$ is an $n \times k$ matrix such that $X := [X_1 X_2]$ i.e $X_1$ consists of the first few columns of $X$ Also, note that: $X^T[I_n - X({X^T}X)^{-1} X^T] = X^T - X^T = 0$ I need this to ...
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1answer
13 views

Prove a transformation is injective if its restrictions are injective.

Suppose that $V$ is a vector space, and let $V → W$ be a linear map. $$V_0 ⊆ V_1 ⊆ · · · ⊆ V_i ⊆ V_{i+1} ⊆ · · · ⊆ V$$ are subspaces of $V$ (one for each $i = 0, 1, 2, \ldots$) and inclusions ...
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3answers
35 views

How do I prove $|\left \langle x,y \right \rangle|=\left \| x \right \|\cdot \left \| y \right \|\Leftrightarrow y=cx,c\in F$

Proving $\Leftarrow$ is easy enough, it's just a matter of plugging it right in. For $\Rightarrow$, I tried changing the right side to $\left (\left \langle x,x \right \rangle \cdot\left \langle y,y ...
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3answers
37 views

Is it possible to solve for values in a matrix such that all rows and columns have equal sum?

Is it possible to solve for values in a grid such that all rows have the same sum and all columns have the same sum where values in the table can be any real number? meaning: ...
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1answer
18 views

Basis of a centralizer

If we consider $A$ to be a $2 \times 2$ matrix, the $2 \times 2$ matrix of ones $J_2=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, and the centralizer $\{A \mid AJ=JA\}$, is there a way to ...
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2answers
57 views

If I know that a matrix $G = (X^{T}X)^{-1}$, how can I recover what $X$ is?

If I have a matrix $G$ where I know that $G=(X^{T}X)^{-1}$, is there a way to find $X$? Specifically, I would like to find $G$ where $G$ is: $$G = \begin{bmatrix} 0.125 & 0 & 0 & 0 & ...
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4answers
34 views

Find the value of c such that the system has a solution other than $(0, 0, 0)$.

I am asked to the find the value $c$ such that the system has a solution other than $(0, 0, 0)$. This is the linear system: \begin{array}{rcrcrcl} cx & & & + & 4z& = & ...
0
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1answer
18 views

exponential of elementary matrix $\exp(tE_{a,b})$

$E_{a,b}$ is the elementary $n\times n$ matrix with $1$ in $(a,b)$-entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a$ not equal to $b$. If $a=b$ then they would be on the diagonal, so ...
0
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0answers
25 views

Possible largest number of column vectors with certain structure in a rank r matrix

My question is: If $A$ is a dimension $p$ symmetric square matrix with rank $r$ ($r<p$), and $a_{ij}$ is the element in the $i$-th row and $j$-th column. How many column vectors can satisfy ...
0
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2answers
42 views

$4$ equations $4$ unknowns by Gaussian Elimination

I've got a question with $4$ points asking me to determine the function: $(-2,-12), (0,3), (2,-1), (4,9)$ I did this the long way by subtracting all the functions and got the answer: ...
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2answers
82 views

How to take the inverse of the matrix $X^{T}X$, when it isn't invertible?

If I have a matrix $X$ and I am trying to compute $(X^{T}X)^{-1}$, which is the inverse of $X^{T}X$. However, each time I try to do it in some computing package like R, I get that $X^{T}X$ is ...
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3answers
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Linear Algebra: Determine if a triangle is a right angled triangle

I've just started my course in Linear Algebra, and I've come across a question I'm not entirely sure how to solve. Let $A = (1, 1, -1), B = (-3, 2, -2), C = (2, 2, -4)$. Prove that the triangle ...
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2answers
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Computing the inverse of $I - \lambda E$ where $E^{k+1} = 0 $ for some $k \geq 1$

If $E$ is a square matrix over $\mathbb{C}$ with $E^{k+1} = 0$ for some $k \geq 1$, then show that $I - \lambda E$ is invertible for all $\lambda \in \mathbb{C}$ by explicitly computing its ...
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3answers
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Show that $\{0\}$ and $V$ are the only linear subspaces of $V = k.$ where $k$ is a field.

On the surface this seemed easy, but my first attempt was rendered useless since I don't actually know that its an ordered field. I said: Let $W (\neq V)$ be a subspace of $ V$ then, if we let $v = ...
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1answer
21 views

How to determine which of the 6 columns of this matrix are not linearly independent when combing with the rest?

I currently have a matrix $G$ with $6$ columns from a simulation that looks like: $$\begin{bmatrix}{} 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 & 0.0 ...
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1answer
22 views

Accelerating linear solve in MATLAB for a specific type of matrices

Inside a DG solver (so far 1D) I need to solve a linear system of equations multiple times. The order of the system is rather small ($N=10..20$). I need to solve the system $Ax=b$, where $A$ is the ...
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1answer
40 views

How to show that the axiom for vector space hold for the following operation?

So the operation is sum defined by $f+'g=f\circ g$ (composite of functions) and usual scalar multiplication. First, for $(x+y)+z=x+(y+z)$ property, $(f+'g)+'h=(f \circ g)\circ h\ne f \circ (g\circ ...
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1answer
35 views

Why is the axiom for vector space not satisfied by the following equation?

Vector sum $(x_1, x_2)+'(y_1, y_2)=(x_1+2y_1, 3x_2-2y_2)$ and the usual scalar multiplication $c(x_1, x_2)=(cx_1, cx_2)$. Sure additive properties does not hold for the operation but why does the ...
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2answers
24 views

Solving this augmented matrix for a linear system

I attempted to do the following exercise with no luck. I was sure of the answer, but apparently not.. somehow... The given matrix is the augmented matrix for a linear system in the variables $x_1$, ...
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0answers
14 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One (x,y) and the other (x',y') as seen in this photo. Coordinate systems I am trying to convert the coordinate in the (x,y) system to the rotated red ...
2
votes
2answers
71 views

Why $x\in\ker A$ implies $x_i-x_j=\lambda \det A_{ij}$?

Suppose that $A$ is a real matrix with $n-2$ linearly independent rows and $n$ columns adding up to $0$. I can show that for any $x=(x_1,\dotsc,x_n)\in\ker A$ (that is, any $x\in\mathbb R^n$ ...
1
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2answers
29 views

Trace of matrix with power $5$

Find trace $A^5$ if $A$=$\pmatrix{1 & -1 \\ 2 & 2} $ I mean to find trace of matrix with any power. but this matrix has imaginary eigen values
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1answer
37 views

$T \circ R=R \circ T$ for Linear Operators

Let $T,R: V \rightarrow V$ be two linear maps of rank 1 with the same kernel and the same image. Prove that $T \circ R=R \circ T$. I cannot use the rank/nullity theorem since $V$ is not necessarily ...
0
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2answers
14 views

Find all values of $k$ such that the system has no solution

I'm trying to solve the following linear system: $x_1-4x_2+4x_3=4$ $4x_2-2x_3=k$ $-x_1+8x_2-6x_3=-1$ This is what I did so far in attempt to solve it (I highly doubt I did it correctly)