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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Determine whether the system $A\mathbb x=\mathbb b$ is consistent by examing how $\mathbb b$ relates to the column vectors of $A$. [closed]

10. For each of the choices of $A$ and $\mathbb b$ that follow, determine whether the system $A\mathbb x=\mathbb b$ is consistent by examing how $\mathbb b$ relates to the column vectors of $A$. ...
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64 views

Curl matrix operation

Consider a vector field $\underline{{f}}:\mathbb{R}^3\rightarrow \mathbb{R}^3$. We know that $\underline{\nabla}\cdot\underline{f} = tr(D\underline{f})$, $D\underline{f} = \begin{pmatrix} \...
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1answer
91 views

Positive power of an invertible matrix with complex entries is diagonalizable only if the matrix itself is diagonalizable.

Show that a positive power of an invertible matrix with complex entries is diagonalizable only if the matrix itself is diagonalizable. The other direction is trivial. This direction seems a little ...
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2answers
40 views

Linear Code Problem, Weight & Minimum Distance

Please the highlighted part in the image below. I don't understand why w(c2) must be larger than s(c1, c2) considering s(c1, c2) is counting the position where c1 + c2 = 0, c1 != 0 and c2 != 0 while w(...
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1answer
46 views

How to find the eigenvalues of a linear operator in general?

My professor assigned us this exercise in class:- Let $V = \mathbb R^{10}$. List five linear maps in $L(V,V)$ (other than the identity map and the zero map) and check if they're diagonizable. ...
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2answers
58 views

Fast projection of multiple points on a line

Assume the following setup, taken from this source. I will omit the arrows from now on. Then the projection of $v$ onto $s$ is given by \begin{align} proj_v = \frac{ \langle v,s \rangle}{\langle s,...
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1answer
35 views

Will this iteration converge to the Left singular vector and right singular vector of Highest singular value?

I am constructing two sequences of vectors $\mathbf{x}_1,\mathbf{x}_2,\dots,$ and $\mathbf{y}_1,\mathbf{y}_2,\dots,$ in the explained manner. All this vectors are unit-norm. Consider the $N\times N$ ...
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1answer
19 views

Finding a subspace

The questions asks: Fix a 3x2 matrix B and let W be the subset of $M_{24}(\mathbb{R})$ which consists of all the 2x4 matrices A such that BA = $0$ (the zero matrix in $M_{34}(\mathbb{R})$). Is W a ...
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34 views

If $u, v, w$ are distinct vectors, $S =\{u, v\}$ is L.I. Prove if $T= \{u, v, w\}$ is L.D then $w \in$ span(S)

I know that $T$ has more vectors than it needs to span its space, that's what it means to be dependent. so I write w i.t.o the first two vectors, $w = \left(-\frac{a_1}{a_3}\right)u - \left(\frac{a_2}{...
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2answers
44 views

Sizing image but need math equation

Here's my issue. I have a website mainly of images, but the image producer creates them to be too big for an average screen size so I'm looking for math equations to use to automatically size them to ...
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1answer
53 views

Langrage Multipliers with two constraints (Efficient Calculation)

Suppose I have to minimize $x^2 + 2y^2 +3z^2$ under $x+y+z = 1$ and $x+2y+3z = k$ where $k$ is a constant and I also require $x,y,z$ to be between $0$ and $1$. I know the traditional way of Lagrange ...
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1answer
32 views

How to find the matrix for this transformation relative to the standard basis

I'm having a lot of trouble with this problem. Any help would be appreciated.
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39 views

Do the eigenvalues and the roots of characteristic function coincide for the matrices with entries in a P.I.D?

I encounter with a problem like this: Let $R$ be a principal ideal domain, A be an upper triangular matrix in $R^{n\times n}$, then the set of eigenvalues of A is same as the set of diagonal entries ...
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71 views

Cross product in uneven matrices

I don't need help with dot product, only the cross product section. Even a hint as to where to start would be great.
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1answer
44 views

Find characteristic, minimal, principal vector and Frobenius form from Jordan form

I am given a Jordan form: $$\left(\begin{array}{rrrrrrr} 2 & 1 & & & & & \\ & 2 & 1 & & & & \\ & & 2 & & & & \\ ...
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0answers
35 views

Estimating parameters of a stochastic matrix

I am stuck with the following problem in research. Let $A_{1}$, $A_{2}$ and $B$ be stochastic matrices. Let $B = f(A_{1},A_{2})$. Let $\pi =[\pi_{1},\pi_{2},\pi_{3}]$ be a vector such that $\sum_{i} ...
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0answers
53 views

How to write this as the sum of two orthogonal vectors?

The system says that my answer is wrong. First I found the projection of $\overrightarrow y$ onto $\overrightarrow u$, which is $\overrightarrow x_1$ and the $\overrightarrow x_2$ is subtracting $\...
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3answers
228 views

Show that $\lambda$=1 as eigenvalue, find one corresponding eigenvector

Here's the question: $\lambda$ The typical formula I've seen is $(A-\lambda I)v = 0$ where A is the starting matrix, $\lambda$ is the eigenvalue, I is: $$\begin{bmatrix} 1&0&0 \\ 0&1&...
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2answers
83 views

On some propreties of orthogonal complements

In my book the following propositions on orthogonal complements are given without any proof. However, I cannot figure out how to prove them, even though they must follow directly from the definition ...
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2answers
115 views

Graph connectedness algorithm idea

Assume we are given a list of edges of a graph. For instance in edge i we are given node numbers a(i) and b(i), being the starting and ending points respectively. I need to write an algorithm to ...
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1answer
46 views

Show that the step function is linearly independent

Consider the set V consisting of all functions $f : \mathbb R \to \mathbb R$, considered as a vector space over $\mathbb R$ with the usual definitions of addition and scalar multiplication. Consider ...
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43 views

Solve the following in vector form:

So i did a substitution to solve the system normally, and got $x=17.67$ $y=9.67$ $z=10.67$ Where I am stuck is how to represent something like this in a vector form, maybe my solution was wrong in ...
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1answer
106 views

Name of the matrix transform $AA^*$ given A?

There are a number of places this matrix transform making its appearance: Every positive semi-definite matrix $B$ can have a decomposition $B=AA^*$ If the matrix $A$ is a lower triangular matrix ...
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Solve the following for $D$: $ABDB^{-1} = I$

So, here's what I know, 'I' is typically: \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} But, from that I really have no idea where to go ...
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Linear operator from matrix to general presentaition

Let $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ be a linear operator defined by: $T(0,1,1)=(2,-1,1), T(2,-1,0)=(1,1,0), T(-1,0,0)=(1,-1,1)$. I need to find $T(x,y,z)$. First of, i have found $[T]_{...
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1answer
91 views

Finding the minimal polynomial of an operator

Let L: V $\rightarrow$ V be a linear operator such that L$^{2}$ + 1$_{V}$ = 0. If V is a real vector space, show that 1$_{V}$ and L are linearly independent and that $\mu_{L}$(t) = t$^{2}$ + 1. My ...
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4answers
134 views

Prove Cauchy-Schwarz equality.

My professor asked me to prove the equality in Cauchy-Schwarz inequality. The equality holds iff the vectors $v$ and $u$ are linearly dependent. I am able to show the equality using the fact $v$ and $...
3
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0answers
125 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
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6answers
95 views

Determine inverse matrix of $\left(\begin{matrix} 0 & 3 \\ 0 & 6 \end{matrix} \right)$ using Gauss-Jordan method

I need to find the inverse of the following matrix with Gauss-Jordan method, but apparently, checking with a calculator, it does not exist: $$\left(\begin{matrix} 0 & 3 \\ 0 & 6 \end{matrix} \...
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1answer
29 views

How to find a basis with 2 constraints

If V is a subspace with $(x_1,x_2,x_3,x_4)\in R^4$ such that $x_1 -2x_2+x_3=0, 2x_1-3x_2+x_3 = 0$ How would I find a basis for this? I cant seem to find a way other than inspection because normally I ...
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1answer
140 views

Finding a basis for a vector space with equations

Let $V = \{ (x_1, x_2, x_3,x_4)\in R^4: x_1-2x_2+x_4 = 0, 2x_1-3x_2+x_3 = 0 \}$ I am trying to find a basis for V. Subtracting the constraints from each other yields $x_4= x_2/2+x_3/2$ this means ...
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2answers
38 views

Linearly independent in different vector spaces

If a set of vectors are linearly independent in $\mathbb{R}^n$, are they also linearly independent in the vector space $V$? Edit: Here is the full question: Let $B = \{v_1,...,v_n\}$ be a basis for a ...
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149 views

Union of two bases in different subspaces question

Not sure about this one. Prove or disprove, If $U$ and $W$ are linear subspaces of a linear space $V$ with the bases $\mathcal{B}_1$ and $\mathcal{B}_2$ ($\mathcal{B}_1$ for $U$, $\mathcal{B}_2$ for $...
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0answers
35 views

Is this tensor identity true?

If We have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ and a symmetric positive definite Matrix $\boldsymbol{M}$ I was wondering if the expression $((\boldsymbol{a}\times \boldsymbol{b}) \cdot \...
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0answers
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Matrix equation $A=PBP^{-1}$

Suppose $P$ is invertible and $A=PBP^{-1}$ . Solve for $B$ in terms of $A$. My attempt: I just left multiplied the equation by $P^{-1}$ and right multiplied it by $P$ so that I got $B=P^{-1}AP$. Is ...
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0answers
33 views

Finding the basis of a vector space $\Bbb{W}$ [duplicate]

Given that $\Bbb{W}=\{(a,b)|a,b\in\Bbb{R}\}$ with addition defined by $(a,b)\oplus(c,d)=(a+c+1, b+d)$ and scalar multiplication defined by $k\odot(a,b)=(ka-k+1, kb)$ is a vector space, find a basis ...
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2answers
38 views

How can I find projection of $x^3$ in $L^2[-1.1]$

In $L^2[-1,1]$ with the usual $L^2$ integral norm, define $$ p_0 = 1, \quad p_1(x) =x, \quad p_2(x) = x^2, \quad p_3(x)=x^3. $$ Let $M$ be the subspace generated by $p_0,p_1,p_2$. Then how can I ...
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3answers
206 views

Linear transformation matrix to find trace

I'm looking for a matrix that it's product with an $n\times n$ matrix that will return the trace of the matrix. Any ideas of how to build one?
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1answer
35 views

What does the pseudo of a non-invertible matrix signify?

More specifically if there is a matrix whose two rows are exactly identically, its inverse can't be calculated because its determinant is 0. However, its pseudo-inverse can still be found out. Is ...
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0answers
59 views

least square solution of overdetermined system with additional unknown

I was hoping somebody could tell me the best way to solve the following overdetermined system for the scalars $x_{1}$,$x_{2}$ and $x_{3}$, where the C $3 \times 1$ vectors are unknown, $A_{i}$ is a $3 ...
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0answers
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antisymmetric matrix in $\mathbb{R}^3$ and cross product

Take $\Omega$ an antisymmetric matrix in $\mathbb{R}^3$ and $D$ a symmetric matrix, Then $\Omega D + D\Omega$ is antisymmetric and since we are in $\mathbb{R}^3$ there exists vector $\theta$ such that ...
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1answer
35 views

linear map combine conditions

Wikipedia states: Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two ...
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0answers
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What is the best way to master my algebra skills without taking an algebra class?

I was in advanced math my entire life. I got through all the math I needed for my original degree. 8 years later here I am changing degrees and I need more math. I just took calculus I and I passed it....
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2answers
87 views

3 Variable System of Equations When All Set to Zero

So I'm doing an a bit of a pre-assessment for something, and I feel like I am missing something on this question: Now I know how to solve a normal 3 variable system, but with this they are all set ...
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1answer
53 views

The lower bound for the smallest eigenvalue given the condition

In a paper, i saw a statement that the smallest eigenvalue of $P$($P$ is reversible Markov chain with stationary distribution $\pi$) is greater than $2 \beta - 1$ with the condition, $P \geq \beta I$. ...
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2answers
60 views

Does matrix representation being same in same basis mean linear operators are same?

Let $V$ be a vector space of dimension $n$. Let $\{i\}=\{i_1,i_2..i_n\}$ be an orthonormal basis for this space. Let $L$ and $L^{'}$ be two linear operators $L:V \to V$ and $L^{'} : V \to V$. Now I ...
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calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
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2answers
28 views

If $\mathcal{B}$ is a basis for a $\Bbb{V}$ and $\Bbb{U} \subseteq \Bbb{V}$ then $\exists\mathcal{S} \subseteq \mathcal{B}$ basis for $\Bbb{U}$

If $\Bbb{U}$ is a subspace of a finite dimensional vector space $\Bbb{V}$ and $\mathcal{B}=\{\vec{v}_1,..,\vec{v}_n\}$ is a basis for $\Bbb{V}$, then some subset of $\mathcal{B}$ is a basis for $\Bbb{...
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1answer
962 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
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1answer
65 views

Intuitive explation for oriented matroids?

Where can you find intuitive explanation on oriented matroids? Other perhaps relevant questions on this How do you get the chirotope of a oriented matroid from the signed circuits? (other than ...