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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proof about rows and columns in linear algebra

I am in an introductory linear algebra course, and I really need help on this question: Prove that if $P$ and $Q$ are $n\times n$ matrices such that at least one of them has rows that don't span ...
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8 views

Solving Principle Component Analysis

Okay guys so i am struggling with theoretical mathematics. So i am given the Principal Component Analysis: Y = $$(X − 1x^T )G$$ where X (n × p) is the data matrix, 1 is a vector of length n ...
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13 views

Linear Transformation from alpha to beta

Hello I am in my Calculus 4 class and I am studying for the final and one thing I've not ever been able to understand is how to do that matrix representation. so I'm working on the practice final ...
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19 views

Finding all orthogonal matrices commuting with a positive-definite matrix

Given $M$ a symmetric positive-definite matrix, I'd like to characterise the orthogonal matrices $Q$ commuting with $M$: $MQ=QM$. $Q$ and $M$ commute if and only if they are simultaneously ...
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1answer
12 views

Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
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2answers
31 views

For what value of k does the following system of linear equations have infinitely many solutions?

I've been struggling for hours trying to solve this: For what value of k does the following system of linear equations have infinitely many solutions? $$x+y+kz=3$$ $$x+ky+z=-7$$ $$kx+y+z=4$$
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1answer
28 views

Kernel and Image of an integral.

Im struggling to answer a question where $F: P_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R}) $ $$F(f)(x)=\int^{x+1}_{2-x} (1-t)f(t) dt$$ So to find the Kernel do i set the integral equal to 0 and ...
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1answer
16 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
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1answer
19 views

Gauss-Jordan elimination/matrix

Hello guys i got a problem from university and i cant seem to find the answer This is the problem : ka+b+c+d=1 a+kb+c+d=1 a+b+kc+d=1 ...
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2answers
35 views

Solution of $A^\top M A=M$ with $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can't find out it there are other such matrices and if so, ...
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2answers
18 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 ...
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2answers
59 views

Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
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2answers
24 views

Nullity of linear transformation

I'm struggling to find the nullity $N(T)$ of the following linear transformation (in the canonical basis of $\mathbb{R^{2\times2}}$ $ M = \begin{bmatrix} 0 & 0 & 0 & 0 ...
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1answer
52 views

Eigenvalues of matrix of order $n+1$

How to find eigenvalues of following matrix? $A=\begin{bmatrix} n & -1 & -1 & \cdots & -1 \\ -1 & 1 & 0 & \cdots & 0 \\ -1 & 0 & 1 & \cdots & 0 \\ ...
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8 views

Convert equation of plane to parametric form , vector form and cartesian form

Find equation of a plane passing through point A(1,2,3), B(3,–1,4), and C(5,1,–4) in: a. Vector form b. Parametric form c. Cartesian form If its equation of line i understand that but for ...
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1answer
15 views

Matlab algorithm for non-orthogonal diagonalization of symmetric matrices

I need to find a basis in which the symmetric bilinear form given by the n x n symmetric matrix which has 2's along the diagonal and 1's everywhere else becomes the identity. That is, if S denotes ...
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1answer
72 views

$\mathbb{Z}[x]$ doesn't have principal maximal ideals

Prove that $\mathbb{Z}[x]$ doesn't have principal maximal ideals. Please, I need help with this problem. Thanks!
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2answers
25 views

Relation between eigenvectors of matrix $X^TX$ and $XX^T$

I found a surprising property of the eigenvectors of the matrix $A = X^T X$ and $B = XX^T$ experimentally. Let $X$ be $n \times d$ with $n > d$. Then $A$ and $B$ are psd matrices. The eigenvalues ...
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1answer
19 views

Basis for the space of 4*4 hermitian matrices with specific anti-commutation properties

The space of 2*2 hermitian matrices can be spanned using the basis involving identity and the three pauli matrices. Here, the pauli matrices have specific properties like: When squared they give ...
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3answers
27 views

A question about linear combination

The question is to show Given a non-zero vector u and a set of non-zero vectors $D=\{v_1,v_2,…,v_n\}$, show that $u$ is not a linear combination of $D$ if $u⋅v_i=0$ for all of $i=1,2,…,n$. It is ...
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2answers
33 views

Check if my trajectory colliding another objects

I'm new to Math.stackechange and i'm a programmer not a mathematician :-(. I'm solving problem in 3D engine for a computer game. But this time i need to do calculations on server side, ...
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19 views

Intersecting 3 Parametric lines

Given $[x,y,z] = [x0,y0,z0] + t[a0,b0,c0]$ $[x,y,z] = [x1,y1,z1] + s[a1,b1,c1]$ $[x,y,z] = [x2,y2,z2] + v[a2,b2,c2]$ How can I solve for the best intersection ...
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1answer
65 views

How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$ f(x) = \frac{1}{2}x^T A x - b^Tx + c $$ My question is: How can we deduce this formula? I understand, ...
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1answer
19 views

Easiest way to compute singular values of matrix

Let $A\in GL_2(\mathbb{R})$ be an invertible matrix. I know $A$ has a singular value decomposition $A=U\Sigma V^T$ where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal. I call "singular ...
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1answer
11 views

how to find the pivot/axis and angle that move one coordinates space to another?

I am writing a plugin for a 3d modeler, and I am stuck. For my plugin, I need to get the axis and the angle used for rotating a 3d object. But I only get the coordinates (~ 3dmatrices) of the objects ...
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0answers
17 views

Consequence of Cramer's rule and Chiò's condensation [on hold]

enter image description hereHi. I can't understand from where we get this property. I think that is a consequence of Cramer's rule or Chiò's condensation but i haven't found any source that talk about ...
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1answer
17 views

signature of the quadratic form: $f(x,y,z) = xy+yz+xz$

I am asked to find the signature of the following quadratic form: $f(x, y, z) = xy+yz+xz$ I have found that matrix wise, $f(x,y,z)= \begin{bmatrix}x&y&z\end{bmatrix}. ...
2
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2answers
344 views

How to find kernel and image?

I have doubts on how to find the kernel and image for this linear transformation $T:\mathbb{C(R)}\rightarrow\mathbb{C(R})$ defined by: $T(f(x))=\frac{f(x)+f(-x)}{2}$, where $\mathbb{C(R)}$ represents ...
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1answer
18 views

3 linearly- independent vectors

Prove or disprove by counter-example: ${v_1,v_2,v_3}$ linearly-dependent $\Rightarrow$ $ {v_1+v_2,v_1+v_3,v_2+v_3}$ are linearly-dependent. tried to find a counter example and couldn't so I tried to ...
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1answer
17 views

How to find this kernel and image?

Good morning. How to find the kernel and the image of $T(p(x))=xp(x)-3p'(x)$ where $p(x)$ is a polynomial of degree less than or equal to n? I'm lost, I know solve homogeneous systems on $\mathbb{R}$, ...
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1answer
15 views

Proving that the line CR passing through intersection of altitudes AP and BQ is orthogonal to AB

How would you go about solving this? I've tried using projections to prove CO.AB = 0 but haven't made much progress.
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28 views

How to identify the closest values multiple of 96?

I've a list of 8820 values spreaded in the interval [0, 1[. Thus, 1/8820 * t, with t ...
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0answers
46 views

Finding three 3x3 Hermitian matrices which anticommute and squares to identity.

How to find three 3x3 matrices which anti-commute and squares to identity? The best method I thought of was to take a general hermitian matrix. Find the constraints(1) on its elements such that it ...
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0answers
24 views

Matrix equation with orthogonal matrix

Is there an orthogonal matrix $\mathbf{B}$ such that, for each ${\mathbf{x}} = {\left( {\begin{array}{*{20}{c}} {{x_i}}& \cdots &{{x_K}} \end{array}} \right)^T},{x_i} \geq 0\;\forall i$, : ...
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0answers
35 views

$\det{\begin{bmatrix}\det A & \det B \\ \det C & \det D\end{bmatrix}}=0$ [duplicate]

Let $A,B,C,D \in M_n(\mathbb{R})$ and let $rank{\begin{bmatrix}A & B \\C & D\end{bmatrix}}=n$. Prove that $\det{\begin{bmatrix}\det A & \det B \\ \det C & \det ...
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Identifying the span of a set of vectors

I'm trying to do a question where I'm asked to 'identify the span of the set $s={[1,-1,2],[-1,1,0]}$, I know this is the linear combination so I considered $a[1,-1,2]+b[-1,1,0]=[a-b,-a+b,2a]$ but I ...
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1answer
31 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
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1answer
30 views

Sum of the entries in the matrix $A^3$

Let $A\neq I$ be a $5\times5$ matrix with real entries such that the sum of the entries in each row of $A$ is $1$. Then the sum of all the entries in $A^3$ is 1)$\space 3$ $\qquad $2)$\space 15$ ...
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1answer
26 views

A multiple choice question on span and linearly independent subset of a vector space.

Let $\{v_1,v_2...v_n\}$ be the linearly independent subset of vector space V, where $n\geq 4$. Set $w_{ij}=v_i-v_j$. Let W be the span of set $\{w_{ij}:1\leq i,j\leq n \}$. Then 1.$\{w_{ij}:1\leq ...
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2answers
46 views

Find the Taylor's Series for $f(x)=x^3-10x^2+6$ about $x_0=3$ [on hold]

Please help me. I want a solution for this question Find the Taylor's Series for $$f(x)=x^3-10x^2+6$$ about $x_0=3$.
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1answer
35 views

Linear algebra proof with exchange theorem

Assume the Exchange Theorem and prove the following: Assume the vector space V is finitely generated. Then there is a natural number n such that the length of a linearly independent sequence is less ...
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3answers
34 views

For what value of k does the following system have a unique solution

So I guess this question states that we know we have a unique solution we just want to know what value of "k" will allow this. Would I even have to take the determinant since I already know there is ...
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0answers
36 views

Linear Transformation vs Matrix

I'm reading Axler's book, and I'm struggling a little bit with the "think in terms of operators" and "compute in terms of matrices". Many times, someone asks a question about matrices, and I'd like to ...
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27 views

Prove that two matrices in $SO(3)$ are conjugate if and only if they have the same trace

The matrix $SO(3)$ is the group of all $3\times 3$ matrices with determinant=+1. I showed that if the trace is equal then they are conjugate but don't know how to show conjugacy implies equivalent ...
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1answer
25 views

How do I draw graph for $x_{1} ^{2}+x_{2}^{2}-2x_{1}-x_{3}=0 $ over a x,y plane? [on hold]

How to draw graph with only 2 coordinates and in the equation there are 3 coordinates?
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1answer
12 views

Prove that orthogonal projections are linear

In chapter 8 of Hoffman and Kunze, an orthogonal projection is defined as follows. Let V be a finite dim inner product space over field K, W a subspace of V. E is an orthogonal projection from V to W ...
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10 views

Change of Coordinates and Basis

Let $P_{2}(\mathbb{R})$ denote the vector space of real polynomial functions of degree less than or equal to two and let $\beta := \{p_{0}, p_{1}, p_{2}\}$ denote the natural basis of ...
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11 views

Finding the span of this mapping in P2

I have managed to do 10 a&b but im stuck on part C, i understand i have to form a span using Po +P1x+P2x^2 but im not sure what to do next, i have an exam later this week and I have also looked ...
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13 views

Linear algebra proof using the exchange theorem

Assume the Exchange Theorem and prove the following: Assume the vector space V is finitely generated. Then there is a natural number n such that the length of a linearly independent sequence is less ...
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2answers
26 views

How to draw graph of $x^{2}+y^{2}-2x-z=0 $ over a plane (x,y)? [on hold]

How to draw this with only two coordinates= $x^{2}+y^{2}-2x-z=0 $