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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Chapter 2 Sec. 2.6 Hoffman Kunze Linear Algebra exercise 1

Let $s<n$ and $A$ an $s \times n$ matrix with entries in the field $F$. Use theorem 4(not its proof) to show that there is a non-zero $X$ in $F^{n \times 1}$ such that $AX=0$. Theorem 4: ...
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842 views

How many linearly independent eigenvectors does this matrix have?

I have a $4\times 4$ matrix which I can write in $2\times 2 $ block form as $$\begin{pmatrix}A&O\\O&B\end{pmatrix}$$ I was asked how many linearly independent eigenvectors it has? I ...
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61 views

operator over inner product

Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$ and let $u,v \in V$ be given. Define a linear operator $u\otimes v: V \rightarrow V$ by $(u\otimes v)x=<v,x>u$, where ...
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65 views

det function in concave

Let $f(A)=(\det(A))^{\frac{1}{n}}$. And assume domain of $f$ is space of positive semi definite symmetric $n\times n$ matrices with real entries. Show that $f$ is concave: $$f((1-t)A+tB)) \ge ...
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112 views

Prove that the determinant of polynomials is zero

Prove that this determinant is zero (this matrix is $n\times n$): $$\begin{vmatrix} f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\ f_2(a_1) & f_2(a_2) & \cdots & f_2(a_n) \\ \vdots ...
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113 views

Upper triangular matrix question

Why is it true that if $v_1,\cdots,v_n$ is a basis of $V$ and $T$ is a linear map with upper triangular matrix with respect to $v_i$ that then $Tv_i \in span(v_1,\cdots,v_{i})$?
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131 views

Calculating Needed Alloy Using Linear System Of Equations

I'm having troubles with this question which involves a linear system of equations. I keep encountering $x$ to be a negative number, which cannot be possible because you can't have a negative number ...
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212 views

linear transformation, ker(T) and im(T) - question from final exam

Assume $T:V\to V$ is a linear transformation, $\mathrm{dim} V = n$. Let $v$ be a vector of $V$ such that for $1\leq k\leq n : v, T(v), \dots , T^{k-1}(v)$ : they are all NOT zero, but $T^k(v) = 0 $. ...
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338 views

How to calculate the number of integral points inside an area

How many integral solutions (x, y) exist satisfying the equation |y| + |x| ≤ 4 My approach: I have made the graph after opening the the modulus in the above equation by making four equations. Now ...
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41 views

linear map property.

I found the following theorem in "Friedberg-Lienar algebra 4ed". " Let $~~V,~W~$ be vector space over field F. Let $~ \varphi ~: V \to W ~~$ be $~~$ isomorphism . Then, For any $~Q \subset V$ ...
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The generalized eigenvectors of linear operator $T$ span space $V$, why?

I'm studying about determinant and I have a problem understanding the following (Proposition 3.4): The problems I have are highlighted with red rectangles. If anyone can, could you clarify these ...
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50 views

Showing a particular map is equivariant with respect to certain group actions

Let $A$ = {triangles in $\mathbb{R^2}$}. We can let $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ be the vertices of the triangle. The group $GL(2,\mathbb{R})$ acts on $A$ by acting on the vectors of the ...
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298 views

Isomorphisms between vector space subspaces

Originally, I was trying to to understand this proof from Axler: Proposition: If V and W are finite dimensional, then $\mathcal{L}$(V,W) is finite dimensional and dim $\mathcal{L}$(V,W) = (dim ...
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71 views

Are there $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$?

I remember an exercise from Roman's Linear Algebra, but now I can't locate it in the book. Anyway, I think it asked to give examples of $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus ...
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192 views

Optimization problem to find an optimal matrix

I need to find a $n\times m$ matrix $N$ with binary values $(0,1)$ which will maximize an objective function. N(i,j)=0 or 1 indicates whether jth offer is made to ith customer $m$ represents number ...
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38 views

If $T:\mathbb{R}^2\to \mathbb{R}^2$ and $T(1, -1)^T=(0,1)^T, \space T(1,1)^T=(1,0)^T$, find $T(1, -7)^T.$

If $T:\mathbb{R}^2\to \mathbb{R}^2$ and $T(1, -1)^T=(0,1)^T, \space T(1,1)^T=(1,0)^T$, find $T(1, -7)^T.$ where $T$ is a linear transformation. I assume I have to use the property of linear ...
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36 views

Finding the Matrix of a linear map $T$.

Let $T\colon \mathcal M_{22}(\Bbb R) \to \mathcal M_{22}(\Bbb R)$ be defined by: $ T\left(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\right) = \begin{bmatrix} 2c & a+ c\\ b-2c & ...
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82 views

Sum of polynomials with no common factors

I have come across this problem in a set of exercises leading to a proof of the Jordan Normal Form. It begins with taking a polynomial $h(x)$ such that $h(L)\equiv 0$ for a linear operator $L$, and ...
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99 views

how to interpret theorem about polynomial factorization over modulo ring?

polynomial $X^n+a_1X^{n-1}+...+a_n \in \Bbb Z_2[X]$ doesn't have linear factors $\iff a_n(1+\sum a_i) \neq 0$. How then $f(X)=X+1$ can has no linear factors? Doesn't the condition expands to ...
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138 views

Finding a point within a 2D triangle

I'm not sure how to approach the following problem and would love some help, thanks! I have a two-dimensional triangle ABC for which I know the cartesian coordinates of points $A$, $B$ and $C$. I am ...
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102 views

This set of matrices whose eigenvalues have non-zero real part is dense

I'm trying to prove the set of the matrices whose eigenvalues have non-zero real part is an dense subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb R^{n^2}$. ...
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33 views

Proving convexity of $X=\{(x,y) \in \mathbb R^2 ; ax + by \le c\}$

Given $a,b,c \in \mathbb R$, how can I prove that $X=\{(x,y) \in \mathbb R^2 ; ax + by \le c\}$ is convex in $\mathbb R^2$? I know that $X$ is convex when $u,v \in X \rightarrow [u,v] \subset X$, ...
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78 views

Given an operator and its' representation in a non-orthogonal basis. Is it normal?

Given T, an operator in $V = \mathbb {C^2}$ and a basis $B = \{ (1,1), (1,0) \}$. Is $T$ a normal operator if $[T]_B = \begin{pmatrix} 1 & i \\ 2 & \frac{1}{2} \end{pmatrix}$ and $[T^{*}]_B ...
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720 views

Determinant and Inverse of a Difference of two matrices

I've got an expression of the form \begin{equation}\det(I-AB)\end{equation} and I'm wondering if there is a way to write this solely in terms of functions of $A$ and $B$. For the particular case I'm ...
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276 views

Which of the following vectors are in ker(T)?

Let T: R2→R2 be the linear operator given by the formula: T(x,y) = (2x-y, -8x+4y) Which of the following vectors are in ker(T)? *Note that ker(T) is the kernel of T. The way I think I should ...
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106 views

How to find a symplectic matrix that satisfies an additional condition

I have problem how to obtain symplectic $4\times 4$ matrix $T$ with one more condition. Matrix $H$ is known and I have it in analytical form, but the problem is how to obtain matrix $T$ which is not ...
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207 views

Eigenvalues of a self-adjoint operator necessarily distinct?

Let's say we have a self-adjoint operator acting on an inner product space (real or complex), represented, of course, by a self-adjoint matrix. I'm looking at the proof for spectral theorem in which ...
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789 views

Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$

I am studying matrix norms. I have read that $\|A\|_{\infty}$ is the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am ...
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145 views

Proving that exists only one basis which is dual to a given basis

Question Let $V$ be a finite dimensional vector space over $\Bbb F$ and $V^*$ it's dual space. Let $f_1 ... f_n$ be a basis for $V^*$. Prove that $\exists ! e_1 ... e_n$ - basis for $V$ s.t. $f_1 ...
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97 views

Solving a linear system with complex eigenvalues

I have the system: \begin{equation} x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x \end{equation} The corresponding characteristic equation is: \begin{equation} \lambda^2-4\lambda+5 \\ \implies ...
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795 views

Finding the inverse of a matrix using elementary matricies

Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that ...
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708 views

Proving a set of linear functionals is a basis for a dual space

I've seen some similar problems on the stackexchange and I want to be sure I am at least approaching this in a way that is sensible. The problem as stated: Let $V= \Bbb R^3$ and define $f_1, f_2, ...
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152 views

Matrix of T, a linear transformation when Im T = Ker T

Let $V$ be a finite dimension vector space, $T:\ V \to V$ a linear transformation, and assume that $\operatorname{\rm Ker} T = \operatorname{\rm Im} T$. Prove that there is a basis $B$ of $V$, so that ...
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56 views

Finding the orthogonal complement of a particular set

Let $\ell^2$ denote the vector space of all square summable sequences with the inner product defined as $\langle x,y\rangle = \sum\limits_{i=1}^{\infty} x_i \bar y_i$, and $\ell_0$ denote the space of ...
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83 views

When A and B are similar matrices,what conditions hold A = C.B

When A and B are similar matrices,In which cases,A = C.B I'm not sure whether this is a valid mathematical question... $A=D^{-1}B D=C B$
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485 views

when is non homogeneous system consistent?

What is the condition for non homogeneous system to be consistent ( single solution or infinite)? I don't know a condition for any solution, when the rank of the matrix equals to the original number ...
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438 views

Given the characteristic and minimal polynomial of a linear operator, find the possibilities of its' Jordan form

Given $T: V \to V$ a linear operator, and given $P_T(x) = x^2(x-1)^5(x+4)^4$ and $M_T(x) = x(x-1)^2(x+4)^3$. Find the possibilities of the Jordan form of $T$. So I said this: Eigenvalues ...
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140 views

more than n eigenvectors

I am learning diagonalization of matrices. We are given the following theorem: If $A$ is an $n\times n $ matrix with $n$ distinct eigenvalues, then $a$ is diagonalizable Now the proof is: ...
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91 views

Complement of invariant subspace

Assuming that I have a given vector space $V$ and a subspace $U$, which is invariant under an endomorphism $A\in End(V)$. I want to prove that $U^\perp$ is also invariant on $A$ .
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102 views

Linear Algebra: homogenous system with only 0 = 0 equations after Gaussian operations

There's a question in Jim Hefferon's Linear algebra book (chapter 1 section 1 problem 3.22), that asks what "happens" when a homogenous system has only 0=0 equations after applying Gaussian ...
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51 views

Length of spanning and independent lists

There is this theorem in my notes that says that in a finite-dimensional vector space any linearly independent list of vectors is shorter than or equal in length to every spanning list. I understand ...
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140 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
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62 views

If $\phi_i$s are linearly dependent, $\det [\phi_i(v_j)] = 0$ - is the proof legit?

Given $v_1, \ldots, v_k \in V$ and $\phi_1, \ldots, \phi_k \in V^*$. If $\phi_1, \ldots, \phi_k \in V^*$ are linearly dependent, proof $\det[\phi_i(v_j)] = 0.$ Here $k$ is the dimension of $V$, but ...
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80 views

A bilinear map's rank

Given a bilinear map $f$, does someone know the terms: the rank of $f$, $\left[f\right]_{E}$ where $E$ is some basis? I tried to find those definitions online and didn't find anything. Thanks!
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581 views

Isomorphism between Hom and tensor product [duplicate]

I am looking for an explicit isomorphism $Hom(V,V^*)\rightarrow V^*\otimes V^*$ where $V$ is a vector space. I thought of: $\phi\rightarrow ((u,v)\rightarrow \phi(u)(v))$ But I'm not sure this ...
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54 views

Jordan form of matrices

So my professor gave me this question: $A=\begin{pmatrix} 0 & 2 & 5\\ -5 & 5 & 10\\ 2 & -2 & -4 \\ \end{pmatrix}$ I had to calculate $\forall 0 < i$ $kerA^{i}$ ...
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280 views

not injective/not surjective linear maps

Let $S$ be the vector space of real sequences, and for $x=(x_1,x_2,\dots)$ define $\alpha(x)=(0,x_1,x_2,\dots)$ and $\beta(x)=(x_2,x_3,\dots)$. The problem was asking for few other things to do, but I ...
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77 views

properties of block matrix

$X=\begin{pmatrix}A&B\\C&D\end{pmatrix}$, $2n\times 2n$ matrix Then a) A, B,C,D are nilpotent ⇔ X is nilpotent b)If X is diagonalisable so is A,B,C,D. c)min polynomial of X divides the ...
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88 views

what is no. of positive eigen value of symmetric matrix A with some given relationship

Suppose A is a 3*3 symmetric matrix s.t. $$\begin{pmatrix} x & y & 1 \\ \end{pmatrix} A \begin{pmatrix} x \\ y\\ 1\end{pmatrix} = xy -1 $$ let p be the no. of positive eigen value of ...
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98 views

Let $A$ and $B$ be $n \times n$ complex matrices. Then

Let $A$ and $B$ be $n \times n$ complex matrices.Then a) $\lim_{k→∞} A^k =0 ⇔$ all eigenvalues of A have absolute value less than $1$ b) $e^A\cdot e^B=e^{(A+B)}$ c) If $A$ and $B$ are nilpotent, ...