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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Aligning matrices, normalization. Calculating coefficients.

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any ...
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16 views

Schur complement of a matrix $A$

Let $A\in\mathbb{R}^{n\times n}$ and its inverse be partitioned $$A = \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end{pmatrix},\:\: A^{-1} = \begin{pmatrix} \tilde{A_{22}} & ...
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18 views

Is it possible to represent {$0, ±m, ±2m, ±3m, \ldots$} in an augmented matrix? [on hold]

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations. Source: Linear Algebra and Its Applications, ...
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3answers
36 views

Find intersection point of two straight lines

I want to find the intersection point of two lines where, one of the lines is parallel to y axis. I know we can find the intersection point of two line by solving the equation $y=m(x-P_x)+P_y$ where m ...
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9 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = ...
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2answers
47 views

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Prove there is a $\lambda$ such that $T^{2}(v) = \lambda T(v)$.

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Let $w$ be a vector from the image of $T$. If $T(w) \neq 0$, prove there is a non-zero number $\lambda$ such that $T^{2}(w) ...
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1answer
55 views

Prove that this $10 \times 10$ matrix is diagonalizable. [on hold]

Suppose that $A$ is an non-invertible $10\times10$ real matrix, and that $\mathrm{rank}(A-3I)=7$ , $\mathrm{rank}(A-I)=4$. How do I prove that $A$ is diagonalizable?
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1answer
24 views

Counterexample of Converse of “rank(PA)=rank(A) if P is invertible”

studying linear algebra , i got a theorem, " Let A be an m x n . If P and Q are invertible m x m and n x n matrices, respectively, then (a) rank(AQ) = rank(A) (b) rank(PA) = rank(A) i know how ...
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1answer
40 views

Does $AA^T = A^TA$ imply that A is normal?

A is $n\times n$ matrix over complex numbers. Does $AA^T = A^TA$ imply that A is normal? If not what will be a counterexample?
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13 views

How to find the irreducible factorisation over Z

So the question is to find the irreducible factorisation of 1-11$\sqrt-2$ over Z[$\sqrt-2$]. I have only been shown how to find this if we have already found the gcd of this with another value, how ...
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1answer
24 views

Finding span of intersection of two vector subspaces

I was trying to follow this answer, but as the comment to that answer suggests, there's a problem with dimensions, and that's exactly where I'm stuck. More concretely, I have subspaces $U$ and $W$, ...
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1answer
12 views

Space generated by vectors

I have a doubt: can you say, for sure, that every space generated by two linear independente vectors with two components generate $\mathbb{R^2}$? For example: $L$ {$(1,1),(0,2)$} = $\mathbb{R^2}$ ...
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47 views

4 points in 3-d space (one known and three unknown)

Problem in 3-d space. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let's name ...
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24 views

How to use Euler's formula to get the following identity

I'm reading a textbook and in the chapter on Euler's formula it is said that it's very useful for deriving all sorts of trigonometric identities, and the example given is: Where ||zθ|| = 1 I've ...
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30 views

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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23 views

Linear Transformation from alpha to beta [on hold]

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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1answer
25 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
18 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
37 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
33 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
27 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
22 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
39 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
21 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
66 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
30 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
59 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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0answers
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
17 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
76 views

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

Prove that $\mathbb{Z}[x]$ doesn't have principal maximal ideals. Please, I need help with this problem. Thanks!
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2answers
30 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
32 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 ...
2
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2answers
36 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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22 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
78 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
22 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
12 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
18 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}. ...
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3answers
364 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
17 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
49 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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25 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
36 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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25 views

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 ...