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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Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then ...
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Are eigenvalues (resp. unit eigenvectors) dependent continuously on elements $a_{ij}$ of a symmetric matrix $A$?

Let $A(t)=(a_{ij}(t)),~(t\in \mathbb R)$ is a symmetric matrix such that $a_{ij}(t)=a_{ji}(t)$ is a real-valued continuous function. Let $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ is all of the ...
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If a subset S of a vector space, V spans V then there exists a subset of S that also spans V. Prove?

Additional related question: Can span of a subset, S of a vector space, V ever be a superset of V. Answer is No! Because V will no longer be a vector space then as V will not be closed under vector ...
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mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
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45 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ ...
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1answer
28 views

Systems of linear equations in the same modulus

I am working with a system of linear equations all taken with the same modulus, $n$, there is no assumption on $n$ other then it is at least 3 (really don't want to assume it is prime) I don't have ...
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44 views

Elegant proof of an elementary result in Linear Algebra

I've been reading Hoffman Kunze, and I came across this theorem (theorem $9$) which has a long and tedious proof. I've been wondering wether there could be a more elegant proof. Theorem 9. Let $e$ ...
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Derivative of the Root of a Positive Matrix [duplicate]

Suppose that the map $t \mapsto A(t)$ from some open subset of $\mathbb{R}$ to the set of positive matrices is differentiable. It is known that the map $t \mapsto \sqrt{A(t)}$ is differentiable, ...
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prove the following results for the eigenvalues of an $n \times n$ matrix $A$ [duplicate]

Prove the following results for the eigenvalues of an $n \times n$ matrix $A$: (a) $0$ is an eigenvalue for $A$ if and only if $A$ is not invertible. (b) $A$ and $A^T$ have the same eigenvalues.
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Linear equation scale transform

I have a linear equation in the general form: Ax + By = C in the standard coordination (Cartesian Coordinate System). I would like to scale this linear's coordination to a custom ratio (for example x ...
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How can one characterise the number of linear combinations of m > 2 linearly independent vectors that map onto the same point in the plane?

I have m > 2 linearly independent vectors v in the plane. I know that two of these vectors are enough to fill the plane. My question is this: How can one characterise the number of linear ...
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16 views

Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: ...
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QR factorization for least squares

This is from my textbook I don't undertand why small errorr in $A^TA$ can lead to large error in cofficient matrix? Because A=QR, so there should be no difference to use A or QR anyway.Could someone ...
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26 views

Prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$

This is the problem I am currently having trouble with: If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix $$ \begin{bmatrix} a & b\\ c & d\\ ...
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1answer
85 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
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[Part of a system of linear equations!]: Find $B$ such that $A = B\times C$, but $C\times C'$ is non-invertable

I have the following Equation: $A = B\times C$ $A$ is a $(N\times 1)$ Known Matrix $B$ is a $(N\times M)$ Unknown Matrix, where $N>M$ $C$ is a $(M\times 1)$ Known Matrix $C\times C'$ is a ...
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Vector norm lemma and proof

I have a question from Numerical linear algebra book by Trefethen & Bau : Let $\|\cdot\|$ denote any norm on $C^m$. The corresponding dual norm $\|\cdot\|'$ is defined by the formula ...
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20 views

Moving vectors to the left and the right of a product

Suppose that $A$ and $B$ are $1\times n$ row vectors and $x$ is a $n\times 1$ column vector. I have an expression $$ (Ax)^2B'B $$ which is an $n\times n$ matrix. Question: Is it possible to write ...
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1k views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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Prove that $\text{rank } T = \operatorname{rank} T^2 \iff \operatorname{Im}T \cap \ker T = \{ \vec 0\}$

$\newcommand{\r}{ \operatorname{rank} } $ Let $T: V\to V$ be a linear transformation with $\dim V< \infty$. Prove that: $$ \r T = \r T^2 \iff \operatorname{Im} T \cap \ker T = \{ \vec 0 \}.$$ ...
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Finding determinant of matrix through row operations [problem help]?

I am having trouble understanding a problem that my Linear Algebra class gave. I understand that determinants can be found through row operations with the following points: 1.) Adding a multiple ...
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Showing a set of vectors is independent

I'd like to redo the proof by walkar (with fewer vectors) just to see if I can handle the algebra the way I do Proving that $(u+v,u+w,v+w)$ is linearly independent Suppose $au + bv = 0 \implies a = ...
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Proof of non-singularity given certain conditions

Suppose that I have a $n\times t$ matrix $\boldsymbol{X}$ that is full rank and a non-singular matrix $\boldsymbol{L} = \begin{bmatrix} \boldsymbol{L}_1 & \boldsymbol{L}_2 \end{bmatrix}$ such that ...
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1answer
2k views

Range of A and null space of the transpose of A

So I'm complete stuck with something. I know it the following statements are true (or at least the seem to be from the results that I got from messing around with it a bit on MATLAB), but I don't ...
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374 views

Let $P$ be the set of all polynomials of degree $\leq 3$ such that $p(t) = t$. Is $P$ a subspace of $P_3$?

Let $P$ be the set of all polynomials of degree $\leq 3$ such that $p(t) = t$. Is $P$ a subspace of $P_3$? I'm not really sure how to solve this. I know that I have to prove that: Since $p, q ...
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43 views

If dim V = dim U and $S \circ T$ is onto, prove or disprove V and W are isomorphic

Let $T: \mathbb V \to \mathbb W$ and $S: \mathbb W \to \mathbb U$ be linear maps. If dim $\mathbb V$= dim $\mathbb U$ and (S o T) is onto (composition), then $\mathbb V$ and $\mathbb W$ are ...
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Tori conformally equivalent

Let $L_1,L_2$ be the lattices generated by $\{1,\tau_1\},\{1,\tau_2\}$ respectively and $X_1=\mathbb C / L_1, X_2= \mathbb C / L_2 $ the corresponding complex tori. We look to prove that $X_1,X_2$ ...
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49 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
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For a linear function the following are equivalent: continuity and Lipschitz continuity

Let $(X,||\cdot ||_X)$ and $(Y,||\cdot ||_Y)$ be normed Vectorspaces over a common field $\Bbb K$. Let $A:X \to Y$ be a linear function. I have to show that the following statements are equivalent: ...
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Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
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Prove that eigenvalues of the operator lie in the interval [on hold]

Let $\phi$ and $\psi$ be two self-adjoint linear operator in Euclidean space, the eigenvalues of which lie respectively in the intervals $[c; d]$ and $[m; n]$. Prove that eigenvalues of the operator ...
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Subtracting scaled projection matrix from identity matrix

I am trying to understand what the following operation signifies. $$ \rm W_n=I-2u_n u_n^H/u_n^Hu_n $$ where I and $u_n$ is described in section 6.3.4.2.3 of this document. My question is, what does ...
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279 views

Direct-sum decomposition of dual space

This is problem #11, Section 6.6 from Hoffman & Kunze, Linear Algebra (p. 213). Let $V$ be a vector space, let $W_1, \ldots, W_k$ be subspaces of $V$, and let $$V_j = W_1 + \cdots + W_{j-1} ...
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Let $S$ be a set of vectors. If each finite subset of $S$ is linearly independent, then $S$ is linearly independent

Here's a similar question which doesn't answer mine: If every subset of $S$ is linearly independent, then $S$ is independent Let $S$ be finite. Let $S_1 \cup S_2 \cup \ldots \cup S_n = S$ such that ...
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3answers
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Proof of Theorem 9.2 of Rudin's Principles of Mathematical Analysis book

I trying to understand a part of the proof of Theorem 9.2 of Rudin's book "Principles of Mathematical Analysis". The theorem says: Let $r$ be a positive integer. If a vector space $X$ is spanned by a ...
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1answer
42 views

Degrees of freedom in a $n \times n$ table

Suppose we have an $n \times n$ table where each row and each column sums to some number $k$. Say that the elements of the table and $k$ are real numbers. Now the question is how many places can we ...
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441 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
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Enlarging a rectangle around its origin, to fit a containing rectangle, but the rectangle must be moved

EDIT: The math was easy/as expected. The bug was in a programming error related to HTML/CSS. Sorry everyone, thank you. I am coding a mobile UI where there is a view of a small card. When clicked, ...
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3answers
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Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
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1answer
21 views

finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
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Can a matrix satisfy all three of the following properties?

Consider an $n \times n$ matrix of the form $$ A = \begin{bmatrix} a_1 & a_2 & \ldots & a_{n-1} & a_n \\ 1 \\ & 1 \\ & & \ddots \\ & & & 1 \end{bmatrix} $$ for ...
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Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?

Assume we have two $n \times n$ real symmetric matrices $ A^2 $ and $B^2$, such that it holds for some $0\leq\rho<1$ $$ 0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2, $$ where "$\leq$" means ...
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1answer
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Elementary reflector $Q$ is orthogonal iff

Recall that an elementary reflector has the form $Q = I + \alpha xx^T\in\mathbb{R}^{n\times n}$ with $\|x\|_{2}\neq 0$. Show that $Q$ is orthogonal iff $$\alpha = \frac{-2}{x^Tx} \ \ \text{or} \ \ ...
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1answer
54 views

Square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue

Prove square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue. My proof: If matrix has eigenvalue z=$\frac{1}{4}(-3+ i \sqrt5)$, then it must has eigenvalue ...
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Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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3answers
33 views

Irreducible polynomials in $\mathbb F_3[x]$

Finding irreducible polynomials in $\mathbb F_3[x]$ of degree less or equal to $4$ for $d=2,3$ the polynomial should not have a root case $d=2$ there are $2\cdot 3\cdot 3=18$ polynomials with ...
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62 views

Proving that eigenvalues are positive iff $\det(A_k)> 0$ for all $k = 1, …, n$ for a real symmetric matrix $A$

I am trying to prove that eigenvalues of $A$ are positive iff $\det(A_k)> 0$ for all $k = 1, ..., n$ for a real symmetric matrix $A$ where $A_k$ is the $k \times k$ matrix obtained by deleting the ...
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1answer
21 views

Prove that the dimension of row space equals to the dimension of column space of an $n\times n$ matrix

Knowing that the row space of $A\in \mathbb{R}^{n\times n}$ equals $N(A)^\perp$ prove that the dimension of column space of a matrix equals its row space dimension. So I'm trying to apply ...
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29 views

Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...