Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Dimension of intersection of two nullspace

Let S and T be two nonzero linear mappings from $\mathbb{R}^n\rightarrow\mathbb{R}$ Prove that $dim(N(S)\cap N(T))\geq n-2$ Note that $N(S)$ stands for nullity of S. I do not have idea how the ...
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Proving if $A$ is an $n\times n$ positive semi-definite matrix, A is Hermitian with non-negative eigenvalues.

I have a test on Monday and the professor hinted that this question might be relevant to the exam, unfortunately, I'm at a loss. As the title states, I would like to prove that if $A$ is an $n\times ...
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How find this maxtrix with the determinant contain a Vandermonde maxtrix

let the matrix $$A=\begin{bmatrix} 1&1&\cdots &1&2\\ 1&2&\cdots&(n-1)&3\\ 1&2^2&\cdots&(n-1)^2&5\\ \vdots&\vdots&\cdots&\cdots&\vdots\\ ...
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Constructing a 2x2 matrix R which represents reflection in the x-y plane,

Construct a 2x2 matrix R which represents reflection in the x-y plane through the line $$(cos(\theta)x+(sin(\theta)y=0$$, where $\theta$ is any real number. (Let's call this line "L".) Write an ...
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1answer
25 views

How to find a basis for the solution space of a linear system?

How to find a basis for the solution space of this linear system? $$ \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ Solutions ...
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22 views

Proving if A is an Hermitian matrix with nonnegative eigenvalues, A is positive semidefinite.

I'm trying to show that if A is an Hermitian matrix with non-negative eigenvalues, then A is positive semi-definite. The only thing I've thought of so far is using Spectral Theorem. I know I want to ...
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1answer
17 views

Orthonormal vector question

It's about question (a), I know how to use the gram schmidt process, however I am missing the third vector in $A$. As it has to be linearly independent can I just choose a random random vector and ...
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45 views

System of Linear Equations Represented by Matrix

Consider the matrix equation: $$ \begin{bmatrix} 2 & 1 \\ 1 & 2 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} = 3\begin{bmatrix} x \\ y \\ \end{bmatrix}$$ I know this might sound ...
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Linear combinations and spanning solution spaces

PLEASE help on part A only :) V = Column vectors [1 0 -1] and [1 3 0]. a) Make a system of 3 unknowns and 3 equations of which he solution space is spanned by V? b) Express [1 2 3] as a linear ...
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how to solve differential equation IVP problems with complex roots? (using matrices)

The problem is: I can find a general solution, but i don't know how to go from there and plug in the Initial values into the equations. i would appreciate any help thank you very much here are ...
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Prove the lines are concurrent (using vectors)

Problem: Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to ...
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28 views

Find the flaw - Sum of two subspaces is a subset of their union

Here is a flawed proof for $V+W\subseteq V\cup W,$ where $V$ and $W$ are subspaces of $\mathbb{R}^n$: Consider $\mathbf{x}\in\left(V\cup W\right)^\perp.$ This implies: ...
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20 views

matrix represented w.r.t. bases satisfying ker and im

Consider the elements $u_1=(1,3)^T$, $u_2=(0,1)^T$ of $^2\mathbb{R}$, and the elements $v_1=(1,2,5)^T$, $v_2=(2,3,0)^T$, $v_3=(0,1,1)^T$ of $^3\mathbb{R}$. You may use without proof the facts that ...
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1answer
29 views

Tensors: traction free planes

Given the following stress tensor matrix, determine the value of $\sigma_{22}$ such that there is a traction free plane and determine the unit normal to this plane. $$ \sigma_{ij} = ...
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Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 ...
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Simultaneously diagonalizable without distinct eigenvalues

It is a well known result that if $u$ and $v$ are two diagonalizable endomorphisms of a $\mathbb{C}$ finite-dimensional linear space $E$, if $u$ (or $v$) has distinct eigenvalues and if $u$ and $v$ ...
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polar decomposition of a square matrix without finding the singular vale decomposition of the matrix.

i am interested in finding the polar decomposition of a square matrix $A = QS$ where $Q$ is orthogonal and $S$ is symmetric and nonnegative definite. i know how to find $S$ first finding the SVD of ...
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1answer
28 views

What is the minimal polynomial of $A^2$?

Let $A\in M_n(\mathbb{C})$. The minimal polynomial of $A$ is $m_A = x^6 - 4x^4+3x^2 +1$. What is the minimal polynomial of $A^2$? I'd be glad for an hint/direction. Thanks!
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1answer
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Setting up a matrix from a recurrence relation to find diagonal matrix?

Considering the recurrence $F_n= F_{n−1}+3F_{n−2}−2F_{n−3}$ where $F_0=0$, $F_1=1$ and $F_2=2$, use diagonalization to find a closed form of the expression. If the sequence is continued the numbers ...
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Plackett-Burman designs for screening experiments

i am looking for a method to investigate the relations between different factors in a experiment. So basically i am looking for a factorial design which fits my needs. I have 4 different lysing ...
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3answers
29 views

Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$ I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$. How can I use ...
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Prove $A$ is scalar matrix

Let $A\in M_n(F)$ and let's assume $A$ has only one eigenvalue. Also, $A$ is diagonalized. Prove that $A$ is a scalar matrix. My Try: $${P^{ - 1}}AP = \left( {\matrix{ \lambda & {} & 0 ...
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The basis for orthogonal complement of a subspace [on hold]

The following is my problem, thank you so much.
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Bilinear form Isomorphsim

Hello I'm trying to give a proof that the following are isomorphic: Bilinar forms and $ T_2^{0}(V) $ Where $ T_0^{2} = F \otimes V^{*} \otimes V^{*} $ and V vector space over F
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2answers
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Proving that $\chi_{T^*}=\overline{\chi_T}$ and $m_{T^*}=\overline{m_T}$ (characteristic and minimal polynomials of adjoint map)?

For a linear map $T:V\to V$ where $V$ is a finite dimensional inner product space over $\mathbb{C}$, I know the result $\chi_{T^*}=\overline{\chi_T}$ (where $T^*$ is the adjoint map for $T$). My ...
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corresponding system of equation of the given solution space

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following: Consider the vectors in $R^4$ defined by ...
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How to applied Gaussian Elimination for non-full rank matrix

I have a question about gaussian elimination. I want to find solution of $$Ax=b$$ as soon as possible using Gaussian Elimination. This is my matrix A ...
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1answer
24 views

Example of matrices with some interesting properties like same characteristic and minimal polynomial etc.

Looking for two matrices $A$ and $B$ with entries in the field $F_2$ with the following properties: $A$ and $B$ both are invertible,have same minimal polynomial,Characteristic polynomial,same ...
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1answer
23 views

Some Dense subset of $M_2(\mathbb{R})$ with its usual topology?

The set of all invertible matrices i.e $GL_2(\mathbb{R})$ The set of all matrcies having both real eigen values. Having $Trace(A)=0$ $3$ is not dense set as It is closed set! $1$ Is dense. take ...
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1answer
31 views

Definition of angle between vectors in spaces with dimensions n

I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my ...
2
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1answer
28 views

Analogy of transpose for a function?

In the page 2 of Linear algebra explained in four pages reference, it has a box describing the relationship between functions and linear transformation. It states that the set of zeroes of a function ...
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Does Least squares solution exist for this case?

$ {\bf{Z}} = {\bf{H}} \cdot {\bf{S}} + {\bf{N}} $ Dimensions of the matrices are as follows: Z = m X m H = m X n S = n X m (matrix S is an orthogonal matrix) N = m X m. All the elements of the ...
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1answer
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Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
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2answers
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Complex projections order in inner product

So the complex projection is defined as $$\operatorname{proj}_\vec{u} \vec{v} = \frac{\langle \vec{v},\vec{u}\rangle}{\langle\vec{u},\vec{u}\rangle} \vec{u}$$ with complex inner product. I was ...
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1answer
40 views

What exactly is Standard Coordinates?

What exactly is a standard coordinates? Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to ...
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establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$. I couldn't understand how to solve it please just give me the first step, maybe I can figure out the ...
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1answer
25 views

If $v\not=0$, then $v/\|v\|$ has norm 1

The question is: Show that if $\vec v$ is a non-zero vector in $\mathbb R^n$ then $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ has norm $1$. I assume that $\vec v=(v_1,v_2,v_3,...,v_n)$ , ...
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1answer
21 views

Verify if symmetric matrices form a subspace

I need to verify if the symmetric matrices form a subspace. But I don't know how to represent a general symmetric matrix. I know that the matrix $A$ is symmetric if $A = A^t$ but I can't write a ...
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1answer
32 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
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1answer
31 views

$C$ & $D$ stuck at calculating

Hello! I am super stuck about how to calculate $C$ and $D$ in the example image, I know it's something simple I just can't figure it out!
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1answer
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How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
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1answer
30 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
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Matrix coordinate blocks and null space of a vector

Suppose we have some non-zero $c \in \mathbb{R}^n$. Let $\{U_i\}_{i=1}^m$ be a collection of $n \times n_i$ matrices where each column is a standard basis vector. Suppose that each standard basis ...
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1answer
39 views

$A^{T}b$ inconsistent system!

I am trying to figure out how the calculation on the last image comes to be (question 9, the yellow area). I have calculated the rest without issue. I know that the formula for the last set is ...
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Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
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2answers
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If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
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26 views

A reduction of Cayley-Hamilton to the complex case [on hold]

I found an interesting proof of the general Cayley-Hamilton theorem but i didn't quite get all of it. I'm trying to make sense of this reduction argument. I'm not that familiar with algebras so this ...
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1answer
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Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
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A linear system for which the solution space is spanned by the given vectors [duplicate]

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
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0answers
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What is recommended for studying Linear Algebra? [duplicate]

I don't know anything about linear algebra and want to start afresh but in a proper mathematical manner. What should I do and which are recommended? I know the basics of multi-variable calculus, some ...