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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I have to show $e_i \in r(T)$ for all $i\neq r$, $1\leq i \leq {n}$.

$D$ be a division ring and $n>2$ a natural number. $e_i$ denotes the element in $D^n$ whose$ (i,j)$-entires are zero. Let $T\in M_{(n-1)\times n}(D)$ such that $T=\begin{pmatrix} T_1 \\ T_2\\ ...
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Calculating the determinant of an interationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$) Could someone explain why this is the ...
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3answers
29 views

To Find the Nullity of a Linear Transformation …

If $V(\Bbb R) $ be the vector space of $2\times2$ matrices and $$M=\begin{pmatrix} 1 & 2\\ 0 & 3 \\ \end{pmatrix}$$ If $T:V(\Bbb R)\to V(\Bbb R)$ be a linear transformation defined by ...
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The relation between the algebraic dimensions of a vector space and its dual

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
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1answer
19 views

What is the relation between the algebraic dimensions of a vector space and its dual?

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
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0answers
16 views

Is it true that $\tilde{P} = D^{\dagger} P D$ has non-negative entries?

Consider a $n \times n$ stochastic matrix $P$ (i.e. non-negative rows sum to one). We are interested in the matrix $\tilde{P} = D^{\dagger} P D$, where $D$ is a $n \times k$ matrix which is ...
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How to express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors?

$u,v$ are linearly independent and $w$ is a non-zero vector. Let $Angle(u,v)=\phi$ and $Angle(u \times v,w)=\theta$. Express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors. ...
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15 views

Which of the following is true for the following linear transformations?

If $T_1$ and $T_2$ are linear transformation on $V_2(\Bbb R)$ by $T_1(a,b)=(0,a)$ and $T_2(a,b)=(a,0)$ , then which of the following is true 1) $T_1T_2=0$ 2)$T_1^2=T_1$ 3)$T_2^2=T_1$ 4)$T_1T_2 $ ...
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30 views

Subspace of $\mathbb{R}^3$: Stuck on closed under addition

$$S=\left \{ \begin{bmatrix} x_{1}\\ x_{2} \\ x_{3} \end{bmatrix} ; x_{1}^{2}+x_{2}^{2}=x_{3}^{2} \right \}$$ Closed under addition: Let $\vec{y}=\begin{bmatrix} y_{1}\\y_{2} \\ y{3} ...
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How to find a set of integer vectors (of length L) such that all its subsets with size L are linearly independent?

Given a number $M\geq L$, how to find a set of $M$ vectors in $\mathbb{Z}_{\geq0}^{L}$, say $S=\{\mathbf{a}_1,\cdots,\mathbf{a}_L\}$, such that: 1-All subsets of $S$ with size $L$ are linearly ...
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Finding ordered basis of a linear transformation so that its matrix representation is diagonal [on hold]

Define $T: M_{n\times n} \to M_{n\times n}$ by $T(A) := A^t$. Note that $T$ is a linear transformation with eigenvalues $1$ and $-1$ with the set of eigenvectors $\{A \in M_{n\times n} \mid A = A^t\}$ ...
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2answers
168 views

Show that 1 and -1 are the only eigenvectors of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
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1answer
23 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
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1answer
25 views

eigenvalues of a matrix $A$ plus $cI$ for some constant $c$

If $A$ is a $n \times n$ real matrix with eigenvalues $\lambda_1,\lambda_2,...\lambda_n$, how does one get the eigenvalues of the matrix $A$ + c$I$, where $I$ is the identity matrix and $c$ is a ...
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56 views

linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
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25 views

Is there any relation Trace and Boundary?

I understand the trace is sum of diagonal elements of a matrix. Further the boundary I always perceive as a 'end points' of bounded domain. However on the link below: ...
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3answers
57 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
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Doubt on Kantorovich inequality. Equivalence of inequalities.

To prove de Kantorovich inequality (for that we suppose the matrix A symmetric and definite positive) I need to demonstrate the next exercise: Proof that $$(x^TAx)(x^TA^{-1}x) \leq \frac{(\lambda_1 ...
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2answers
23 views

Orthogonality and projections

1)Consider the vector space $\mathbb{R}^n$ with usual inner product. And let S the subspace generated by $u\in \mathbb{R}^n,u\neq 0$. Find the orthogonal projection matrix $P$ onto the subspace ...
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24 views

How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
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1answer
17 views

Finding a matrix by using hermitian

$A=\left[ \begin{array}{ccc} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i ...
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Why does $\|A\|^2_2 \geqslant \|Av^k\|^2_2$ where $\lambda_k$ is the largest eigenvalue of $A^TA$

Could someone explain why: Let $\lambda_k$ be the largest eigenvalue of $A^TA$, then $$\|A\|^2_2 \geqslant \|Av^k\|^2_2$$ ($v^k$ is the eigenvector corresponding to $\lambda_k$) From a ...
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2answers
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Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A ...
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53 views

For what values of $h$ the following system is consistent? [on hold]

For what values of $h$ the following system is consistent? $$ \left\{ \begin{array} 3x_1+4x_2-8x_3=h, \\ -6x_1-5x_2=2, \\ x_1+x_2-x_3=1. \end{array} \right. $$
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5answers
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Proving any vector in $\Bbb R^n$ can be written on the form $x = u + v$

I'm having a hard time understanding the solution of this exercise. The exercise says: Let A be an $n\times n$ matrix so that $$A^2 = A$$ Show that every vector $x$ in $\Bbb R^n$ can be written as ...
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2answers
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Let $C(X)$ be the column space of $X \in M_{n \times p}$. Starting off the proof of $C(X) \cup C(X)^{\perp} = \mathbb{R}^n$

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Prove or disprove the following statement: Every vector in $\mathbb{R}^n$ is in either $C(X)$ or $C(X)^{\perp}$ or both. I interpret ...
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111 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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34 views

Is set of three linear equations with three unknown solvable?

I have the following set of linear equations with the unknowns $h, n, i$ which I would like to express as a function of my known quantities, $e, f, g$: $$ e = h - n\\ f = h - i\\ g = i -n $$ with ...
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How can you find a matrix given you know its kernel/nullspace?

Suppose we are given that $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and also that $\ker\phi$ is the span of $\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 ...
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Linear Operator with finite dimension

I'm involved with this exercise. I would greatly appreciate your help Let $V$ be a vector space of dimension $n$ over a field $F$. Let $T: V \rightarrow V$ a linear transformation whose image and ...
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1answer
22 views

Determining whether a set is linearly independent.

I am currently trying to determine whether the following set is linearly independent: $u=(4,3,-2), v=(2,-6,7), w=(14,-12,17)$ It can be easily observed that $w=2u+3v$ and since w can be expressed in ...
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Same eigen values giving 2 different eigen vectors

For the matrix below I am getting two eigen vectors for a single eigen value $$ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ ...
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2answers
136 views

How to determine whether a set is a vector space or not?

I'm currently learning Vector Spaces and although I understand the definition of what a vector space is, I can't seem to be able to find the correct answers when doing some questions. I would even say ...
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1answer
21 views

all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
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Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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2answers
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Injective linear mapping maps every plane to a plane through the origin?

Why an injective linear mapping from $R^3 \to R^3$ maps every plane to a plane through the origin? I can not understand this. It says also that if the mapping is not injective, it maps some line to a ...
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0answers
38 views

Matrix problem, subspace

Suppose you are given a matrix A and have calculated an echelon form R of A. (Note: R is not assumed to be in reduced row echelon form.) Which of the following statements must be true? (Select all ...
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1answer
32 views

Is inequality $tr(A^{-1^T} B) tr(A^T B^{-1}) \leq constant$ correct?

I have the following optimization problem \begin{align} \min_{A} &tr(A^{-1^T} B)\cr \text{subject to} &x^T A x > 0 \cr & A_{ii}=1 \end{align} where $A$ and $B$ are some positive ...
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2answers
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Is the following set of vectors in $\Bbb R^3$ linearly dependent?

I am using Anton's Elementary Linear Algebra book (8e) and trying to do exercise set 5.3, question 2a It gives the vectors $(4,-1,2)$, $(-4,10,2)$ and asks if they are linearly dependent . My final ...
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1answer
28 views

For a given matrix $X$, find two linearly independent vectors in $C(X)^{\perp}$.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ Is there an easy way to ...
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1answer
47 views

Showing a $2\times2$ matrix is invertible

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that ${A} {w} = {v}.$ Show that ${A}$ is invertible. I have no idea on how ...
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2answers
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Find an orthogonal basis for the space spanned by the columns of the given matrix.

Let $$X = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 1 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 1 \\ 1 & 6 & 4 \end{pmatrix}$$ It is immediately clear to me ...
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3answers
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To prove that on $C([0,1])$, the integral $\int_{0}^{1} f(x)g(x)dx $ defines a scalar product.

Now, for the given operation to be a scalar product, I know I need to check four conditions. Here's what I have done so far: $\langle f,g\rangle $ = $\int_{0}^{1} f(x)g(x)dx$ = $\int_{0}^{1} ...
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1answer
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Find the matrix relative to the standard bases

Let $T:P_2 \to P_3$ be the linear transformation defined by $T(p(x)) = xp(x).$ Find the matrix for $T$ relative to the standard bases $B = \{u_1, u_2, u_3\}$, $B' = \{v_1, v_2, v_3, v_4\}$. $u_1 = ...
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1answer
51 views

Basis and dimension of the span of the vectors (0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)

Find a basis for the given subspace by deleting linearly dependent vectors. $S = \text{span}\{(0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)\}$ I do not understand how to "delete linearly independent ...
3
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1answer
40 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. I know that $c$ is an egenvalue and ...
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1answer
43 views

About summer course or online course of Linear algebra and real anyasis [on hold]

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
1
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1answer
25 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
1
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1answer
17 views

Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
0
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1answer
34 views

Questions about Eigenspace

I'm learning about Eigenspaces and have a few questions. Do eigenspaces, eigenvalues, and eigenvectors correspond to a tranformation or can a single vector space $V$ have an eigen-stuff? Is an ...