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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On Matrix polar decomposition and absolute value operator

The polar decomposition for complex matrices is $A=OP$ where $O$ is a partial isometry and $P$ is (hermitian) positive semidefinite. In other notations the matrix P is considered as an absolute value ...
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linear and a projection

Let X be the space of continuous functions f in the interval $[-1,1]$. Define Pf to be the even part of $f$, that is $(Pf)(x)= \frac{f(x)+f(-x)}{2}$ Prove that P defined above is a projection. I'm ...
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48 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
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Prove L*L is invertible if and only if kerL = 0.

Let $L:V \rightarrow W$ and $L^*: W \rightarrow V$, and $L^*L:V \rightarrow V$ on $V$. Prove Lemma 39.3: (a) All eigenvalues of $L^*L$ are real and non negative. (b) $L^*L$ is invertible if and ...
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Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
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Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
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Least squares of symmetric positive semidefinite matrices

What's the best (in terms of computation time and numerical robustness) way to find the least squares solution of $$Ax = b$$ if $A$ is symmetric and positive semi-definite? If $A$ were symmetric and ...
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17 views

Defining operations for a vector space

I was hoping someone could help me with the following. Is it possible to define operations + and $\cdot$ on this set to make it a vector space: \begin{equation*} ...
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1answer
16 views

The difference between norm and modulus

I'd like to know the difference between norm of a vector, ||v|| and the modulus of a vector, |v|
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51 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.
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6 views

pseudoinverse of vec-transpose operator

I'm struggling to find closed form solution for the Moore-Penrose pseudoinverse of the following singular matrix: $$ P + I $$ where P is a vec-transpose operator matrix, defined by: $$P=\sum_{ij} ...
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Mirror a line over a plane

I am trying to mirror a line over a plane, but I am not sure if I am doing it right, so please tell me if something that I do is wrong. I have 2 points $A(1, 2, 1);B(-1,0,2)$ and I have to mirror the ...
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2answers
41 views

If T(S) is linearly independent, show S is linearly independent

Let $T: V \to W$ be a linear transformation. Let $S = \{v_1,...,v_k\}$ and assume $T(S)$ is linearly independent. Show S is also linearly independent. I think I just have to prove that if $a_1 v_1 + ...
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1answer
21 views

Prove that Orthogonal Set Is Linearly Independent

Suppose that $V$ is an inner-product space; $(\space ,\space )$ is our inner-product. I have seen many proofs that go as follows: Let $\{x_1, x_2 ,\ldots, x_n\}$ be orthogonal. Set $a_1x_1 + a_2x_2 ...
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26 views

regu tools l_curve regularization stanford ee 263

I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...
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1answer
10 views

Finding a basis for set of vectors (columns/rows)

I was wondering if someone could help me with the following. I have to find a basis for the subset of $\mathbb{R}^4$ spanned by $(1,2,0,3)^T$, $(3,5,1,7)^T$, $ (1,1,1,1)^T$ and $(0,1,-1,2)^T$. Now I ...
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2answers
26 views

Linear operators and matrices

Let $B=\{(1,0),(1,1)\}$ be a basis of $\mathbb R^2$. Given the following matrix representation of an linear operator $T$ over the basis $B$: $$[T]_B= \begin{pmatrix} -1 & 1 \\ ...
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23 views

$A$ is an $n \times n$ invertible matrix, prove that $f(\mathbf u, \mathbf v)= \mathbf u^TAA^T \mathbf v$ defines an inner product on $\mathbb R^n$

I have difficulty especially proving that $f(\mathbf v, \mathbf v) \geq 0$ for all $\mathbf v$. Thanks
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2answers
38 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
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113 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
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10 views

how to find find a matrix by the characteristic vlaues and vectors

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen values) and characteristic vectors (eigen vectors). But my question is why cant we ...
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1answer
28 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
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2answers
29 views

Uniqueness of Singular Values

Given a matrix A, one inductively constructs (and thus proving its very existence) the singular value decomposition as follows: take $ \sigma_{1}=||A||_{2} $, and consider a couple of vectors such ...
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24 views

Geometrical interpretation of the following transformation

I am learning linear algebra from Linear Algebra by Hadely and I came across this question that I do not have any idea how to solve Interpret geometrically the transformation produced on $E^2$ by ...
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11 views

Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
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19 views

Show a set is in the null space of the transpose of A

I'm trying to show that for $A \in F^{MxN}$ (a matrix with the nth column $a_n$) the following set is in the null space of $A^T$, that is: $N(A^T) = \{x \in \Re^M : A^Tx = 0\} = \{x \in F^M : ...
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2answers
57 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
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2answers
41 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
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1answer
17 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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19 views

time and work problem

A man can begin a work at his maximum rate; but afterwards the rate at which he works follows a cyclic pattern. Every 2 hr,it reduces by half but after 8 hrs,it comes back to its maximum level.He can ...
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31 views

Vector space clarification

I'm asked to decide if the following are vector spaces. A=$\{f:[0,1] \to \mathbb{R}:\int_0^1|f(x)|dx=0$ $\}$ B= $\{f:[0,1] \to \mathbb{R}:f'(x)+4f(x)=0$ and $f(0)=1 $} C=$\{f:[0,1] \to ...
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44 views

Eigenvalue of linear operator [on hold]

Let $V=P_3(\mathbb{R})$ and $T$ is a linear operator such that $T(f(x))=xf'(x)+f''(x)-f(2)$. Find the eigenvalues for $T$ and an ordered basis such that $[T]_B$ is a diagonal matrix.
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34 views

Maps from $SO(3)$ to $S_1, S_2$, and $S_1 \times S_2$

I am looking for continuous maps between the special orthogonal group of 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
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27 views

Eigenvalues and eigenvectors of Linear Operator?

Find the eigenvalues of $T$ and an ordered basis $B$ such that $[T]_B$ is a diagonal matrix. $$ T(a,b)=(-2a+3b, -10a+9b). $$ I know how to find eigenvalues of matrices, but the linear operator ...
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1answer
33 views

Why is the adjoint a useful concept

I've been reading my linear algebra book, and am now on the section about the adjoint of a linear operator, I get the definition provided and even think I understand the general proof of existence and ...
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1answer
45 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
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A question on basis of vectorspaces and subspaces

Let $V$ be a finite dimensional vector space and $W$ be any subspace . It is known that if $A$ is any basis of $W$ then by "extension-theorem" , there is a basis $A'$ of $V$ such that $A \subseteq ...
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
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3answers
28 views

You can only take the span of linearly independent vectors?

Ok, this might be a bit trivial but I'm having trouble wrapping my head around my text book. So, to my understanding for ${Span(v_{1},v_{2},..,v_{n})}$ then ${v_{1},v_{2},..,v_{n}}$ must be linearly ...
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1answer
27 views

Matrix representation of linear operator?

$T$ is defined by $T(p(x))=p'(x)$ $B$ is the basis $\{1,x \}$ Now the book is telling me that $[T]_B$ is the matrix $$ \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} \right] $$ why ...
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1answer
21 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
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Iteratively find solutions for $AB=H$, where $A$ is known and invertible and $B\geq0$ and $H\geq0$ is unknown?

Iteratively find solutions for $AB=H$, where $A$ is known and invertible and $B\geq0$ and $H\geq0$ is unknown? Can we find one solution by the following procedure? First random pick $H$, then $B$ ...
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How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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1answer
25 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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20 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...
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Analogy between linear basis and prime factoring

I recall learning that we can define linear systems such that any vector in the system can be represented as a weighted sum of basis vectors, as long as we have 'suitable' definitions for addition and ...
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1answer
33 views

Find values so matrix not invertible?

$$ \begin{pmatrix} 2 & 4 & k \\ 1 & 3 & 2 \\ 3 & k & 9 \\ \end{pmatrix} $$ For what values of $k$ is the above matrix not invertible. Need help. Don't know where to ...
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How are specific linear maps defined?

I'm revising for exams and a question that often crops up is: given a linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$, describe how to represent $\mathcal{T}$ as a matrix relative to bases ...
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Linear Algebra: show $\sum_{m=1}^{M} a_m x_m = 0$ is a subspace

I have a problem that I can't get my head around. It says that a is any vector in $\mathbb{F}^M$ and to verify (by the three properties of subspaces) that $\sum_{m=1}^{M} a_{m}x_{m} =0$ is a subspace ...
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22 views

Vector Space confusion

For each of the following, I need to decide if it is a vector space over $\mathbb{R}$. (You may assume that the set of all real valued functions on the interval $[-1, 1]$ is a vector space with the ...