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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Is it true that $u + v$ is an eigenvector corresponding to the eigenvalue $\lambda$?

Let $A$ be an $n \times n$ matrix, and $u, v$ be eigenvectors corresponding to an eigenvalue $\lambda$ of $ A$ (that is, $Au = \lambda u$ and $Av = \lambda v$). Is it true that $u + v$ is an ...
0
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1answer
31 views

Isometry in a finite dimensional vector space is always surjective

My book defines an isometry as a linear operator between two vector spaces X and Y where: $$\|T(x)\|=\|x\|$$ Later it has a sentence which I do not understand. If we have a finite dimensional ...
0
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2answers
22 views

Largest eigenvalue of a block diagonal matrix is an eigenvalue of the largest block?

Consider this square matrix $C = \begin{bmatrix} A& 0 \\ 0 &B \end{bmatrix}$, where $A$ and $B$ are also square matrices. Suppose $A$ is larger in the sense that is an $n \times n$ matrix, ...
0
votes
4answers
32 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
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0answers
34 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
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0answers
21 views

How to find the rank of a toeplitz matrix?

Is there any trick to compute or estimate the rank of a toeplitz matrix ? Or is this still unknown for a general toeplitz matrix ?
2
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2answers
277 views

Find the matrix A of the linear transformation T(M)

I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though. I was looking at this ...
0
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3answers
78 views

Orthogonal diagonalization of Symmetic Matrices

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial \Delta (t). Step 2: find the eigenvalues of A which are the roots of \Delta (t). Step 3: for each ...
1
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1answer
29 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
2
votes
1answer
47 views

Let $a,b \in R^5$ be two independent vectors and let $W$ be the subspace spanned by $a,b$. Prove that $U\cap W$ contains a non-trivial vector

Let $a,b \in \Bbb R^5$ be two independent vectors and let $W$ be the subspace spanned by $a,b$. Let $U=\{(x_1, x_2, x_3, x_4, x_5) \in \Bbb R^5\colon x_1 + 2x_2 + x_3 - 3x_4 + x_5 = 0 \}$. Prove ...
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0answers
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Is the steinitz exchange lemma necessary to establish invariance of 'basis-size'?

I am going to answer my own question in some sense... In Beardon's "Algebra and Geometry" he proves (Theorem 7.2.2) that if $v_1,\ldots,v_n$ and $u_1,\ldots,u_m$ are both bases for some $F$-vector ...
0
votes
1answer
13 views

Minimal polynomial of a matrix whose elements have a certain form

Find the minimal polynomial of the $n$-dimensional matrix $(a_{ij})$ when the matrix elements $a_{ij}$ have the form $a_{ij} = u_i v_j.$ Let $A=uv^T$ where $u,v$ are column vectors. Then ...
2
votes
1answer
39 views

Method of orthogonalization that preserves invertibility

Is there a method of orthogonalization such that, given an invertible matrix $A$ with entries in the real numbers, applying the method and then inverting the result is the same thing as applying the ...
2
votes
1answer
58 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
3
votes
2answers
41 views

Pairwise commuting nilpotent matrices: alternative solution needed

I have a problem: Let $A_1,A_2,...,A_n$ be $n\times n$ nilpotent matrices which are commute in each pair ($A_iA_j=A_jA_i$). Prove that: $$A_1A_2...A_n=0$$ I have got a solution by proving that ...
0
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1answer
75 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
1
vote
1answer
27 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
0
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4answers
38 views

Prove that T = I with Linear Transformations.

Suppose that $T \in L(V)$ and $T^2 = I$ and -1 is not an eigenvalue of T. Prove that T = I. What I tried was: Suppose $\lambda$ is an eigenvalue of T such that $T(v) = \lambda v$ Then we know that ...
0
votes
1answer
15 views

LLL algorithm in pari/gp

I know that in PARI/GP the function qflll performs LLL algorithm on a set of bases. However, is it possible for me to look at the code for ...
1
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1answer
126 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
2
votes
1answer
364 views

Reflection Matrix linear algebra

I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states: If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a ...
1
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1answer
231 views

prove that T-cyclic subspace of V generated by x is T-invariant

Let $T$ be a linear operator on a vector space $V$, and let $x$ be a non-zero vector in $V$. The subspace, $$W = \operatorname{span}(\{x,T(x),T^2(x),\ldots\})$$ I have to prove that $W$ is a ...
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2answers
40 views

Fourier coefficients with respect to an orthonormal basis for an inner product space

$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$. Apply the Gram–Schmidt process to the given subset $S$ of the inner product space $V$ ...
0
votes
1answer
19 views

How do you get nullspace N(A) to be orthogonal to C(A^H)

In the picture below, C(A) is given in number7, but I am doing number_8. Ii did a gauss jordan where by i subtracted R2-iR1 to get 0 belo 1st pivot and 1 as the second pivot in column2, row2. Then I ...
2
votes
2answers
90 views
+50

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
3
votes
1answer
64 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
0
votes
2answers
256 views

Orthogonal complement of the diagonal matrices in the inner product space of matrices

$V$ is the matrices space (scalar over the complex). definition of inner product space is: $(A,B)=tr(AB^*)$. $A$,$B$ matrices. assuming $D$ is the subspace of all Diagonal matrices. I need to find ...
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0answers
24 views

Calculating dimension of subspace of all commutating matrices with fixed matrix A . [duplicate]

$A$ is a given point in the vector space $M_{n\times n}(F)$ on the filed F. define the subspace $$W_A=\{B\in M_{n\times n}(F) : AB=BA\}$$ of $M_{n\times n}(F)$ . The question is this that what is ...
0
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0answers
27 views

Proof that quantum relative entropy is $\leq$ 0 using Klein's inequality for positive semi-definite operators

I was asked to prove that $S(\rho) \leq - {\rm Tr} \left[ \rho \log \tau \right] $ where $\rho, \tau$ are density operators on a finite dimensional complex inner product space and $S(\rho)$ is the von ...
6
votes
3answers
237 views

Properties of trace $0$ matrices: similarity, invertibility, relation to commutators

$1.$ Are trace $0$ matrices always of the form $AB-BA$? $2.$ Is a trace $0$ matrix over the complex field always similar to a matrix with $0$ as a diagonal element? $3.$ Is a trace $0$ matrices over ...
0
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2answers
49 views

Characteristic polynomial and eigenvalues of a $3 \times3$ matrix.

Hi so I have to find the characteristic polynomials and the eigenvalues of the matrix: $$A = \begin{bmatrix}1 & 0 & 3\\2 & -2 & 2\\3 & 0 & 1\end{bmatrix}$$ So I know you use ...
4
votes
2answers
60 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
1answer
26 views

Determining if Linear transformation

Please help me get and understand this concept of linear algebra based on this questions: Determine whether or not $T$ is a linear transformation from $\Re^2$ to $\Re^2$ if u$ \in \Re^2 $ and v$ \in ...
2
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2answers
26 views

The inverse of a linear transformation $A$ can be expressed as a polynomial in $A$

Suppose that $A$ is a non-singular linear transformation of an $n$-dimensional linear space into itself. Show that there exists some polynomial $c_0+c_1z+\ldots+c_kz^k$ so that ...
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0answers
22 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
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2answers
38 views

Why use Gauss Jordan Elimination instead of Gaussian Elimination, Differences

Why use Gaussian Elimination instead of Gauss Jordan Elimination and vice versa for solving systems of linear equations? What are the differences, benefits of each, etc.? I've just been solving ...
0
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0answers
33 views

What is the linear combination of B?

I have a problem where I am finding $A^n$B where B=$[3,1,1]^t$. I know the steps in solving, but I do not remember how to find linear combination. I do not see it. There has to be a way to calculate ...
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0answers
11 views

Trouble understanding Hoffman / Kunze exercise [duplicate]

I am finishing up a number theory class and will be studying graduate Linear Algebra in the fall so I thought I'd start early getting familiar with the text and authors by doing some of the early text ...
1
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1answer
21 views

Particular solution to the matrix form Ax=b

This question is more for general understanding than looking for a specific answer. I have a theorem that states: "If Ax=b has a solution x$_p$, then the general solution to the equation is x$_p$ + ...
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5answers
123 views

Definition of determinant [closed]

Determinant is a certain function from the set of all $n\times n$ matrices to the set of scalars. How is the determinant defined? What characterizes the determinant function?
4
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1answer
43 views

Geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts

What's the geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts? For example, a skew-symmetric matrix on its own can be interpreted as an infinitesimal rotation. As ...
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1answer
27 views

Prove existence of Diagonalizable Matrix

Suppose R, T $\in L(F^3)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S $\in L(F^3)$ such that $R=S^{-1}TS$. What I got so far is that since R and T have three ...
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3answers
67 views

Proof that $\mathrm{ker}(A^{T}A) = \mathrm{ker}(A)$?

Is there a proof that can help me understand why this is the case? I can't conceptualize the reason for this in my mind. Thanks.
3
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5answers
578 views

Are inverse matrices unique?

Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that $A,B \text{ have the same inverse matrix} \iff A=B$?
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2answers
61 views

If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
0
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2answers
48 views

Finite dimensional subspaces of inner product spaces are orthogonally complemented

Can someone please explain the proof of the theorem below? I've been looking at it for hours and couldn't figure out how to prove it. Thanks! Suppose $U$ is a finite-dimensional subspace of $V$. ...
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1answer
48 views

Eigenvectors for normal operators and their adjoints

Can someone tell me if this proof is correct? Claim:V is a vector space over the Complex field. $T:V\rightarrow V$ is a normal operator. Then if $v\in V$ is an eigenvector with the eigenvalue ...
2
votes
1answer
50 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
0
votes
1answer
51 views

If there is the inverse operator of the operator A, then $(A^{-1})^{-1}=A$?

A friend of mine asked me today for this example: If there is the inverse operator of the operator A, then $(A^{-1})^{-1}=A$? But I do not have the ability to help, so I told him that his example ...
0
votes
2answers
37 views

Linear Programming with 3 variables

What is the optimal solution for maximizing $X_1 + 2X_2 + 3X_3$ subject to the constraints that: $X_1 + X_2 + X_3 \leq 9$, $-X_1 + 2X_2 + 5X_3 \leq 15$, $X_1 \geq 0$ $X_2 \geq 0$. My answer is ...