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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Gram Schmidt orthogonalisation problem

I was trying to solve some problems related to Gram-Schmidt orthogonalisation when I came across this question Use Gram Schmidt process to obtain an orthonormal set of vectors from the vectors ...
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Sylvester domains

I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved ...
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Determine if all vectors of the form (a,b,c), where b=a+c+1 are subspaces of R^3?

Determine if all vectors of the form $(a,b,c)$, where $b=a+c+1$ are subspaces of $\mathbb{R}^3$? Use the theorem: If $W$ is a set of one or more vectors from a vector space $V$, then $W$ is a ...
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59 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
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$XD = DX$ then X is diagonal

Let $D$ be an $nxn$ matrix with diagonal entries $d_1, ..., d_n$ all distinct. Prove that if $XD = DX$ then $X$ is diagonal. Not sure how to approach this, but here is my attempt, denote ...
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A question regarding linear independence.

A problem in a textbook goes as follows: Show that if S is a linearly independent set of vectors, then so is every nonempty subset of S. Is it acceptable to say that since every vector in S is ...
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Finding complex solution to $X^2 = A$

I'm having troubles with coming up with a solution to part (iii), an idea was to use: $$ X^2 = A \iff P^{-1}X^2P = P^{-1}AP = D$$ and note that $P^{-1}X^2P = (P^{-1}XP)^2$, so we can solve $Y^2 = ...
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Bessel-(LIKE) Inequality

Let $H$ be the Hilbert space, and let $M_1,M_2,...,M_n$ be mutually orthogonal closed linear subspaces of $H$. If $P_{M_i}x=x_i$, then show that $$\sum\limits_{i=1}^n\|x_i\|^2\leq\|x\|^2 ,$$ The ...
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Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
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Basic question from LA: Why do we find Roots of a polynomial?

This may sound like a basic question, but I am sorry to say that I did not find it's answer which completely satisfy my query. Here is the question: "What is the need to find roots of a polynomial ?" ...
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39 views

How can I prove this formula is nonsingular?

I have a formula $$A^{\mathrm T}+QA^{-1}G$$ where $A$ is nonsingular, $Q$ is positive semi-definite, $G$ is positive semi-definite, $(A,G)$ is controllable, and $(Q,A)$ is observable. My question is ...
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How many elements does $\mathbb{Z}_{p^{\alpha}}^*$ have?

Let $p\in\mathbb{N}, p>2$ be a prime number and $\alpha\in\mathbb{N},\alpha\geqslant 2$. How many elements does $\mathbb{Z}_{p^{\alpha}}^*$ have? What I know is that $$ ...
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Simple graph equation help

I am trying to take a mock exam and this is one of the questions: It is asking me for the equation of the graph. It gives me the following options: ...
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How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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puzzle on [13,10,3] perfect Hamming code over $\mathbb F_{3}$

The soccer betting form contains a list of 13 games. There are three possible outcomes for each game: “the first team won”, “the second team won” and “draw”. Each betting form allows to chose one ...
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48 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
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Sum of matrices

Let $A$ be an $m×n$ matrix and $B$ be an $n×p$ matrix , then how do we show that $AB$ can be written as a sum of $n$ matrices each of order at most $1$ ?
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V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
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1answer
65 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
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Is $hh^T$ positive semi-definite ($h$ is a column non-negative vector)? [duplicate]

Is $hh^T$ positive semi-definite? It seems to be positive semi-definite, but I cannot prove it. Please help:)
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Prove Linear Dependence in T: V -> W

Problem: "Let $V$ and $W$ be vector spaces and let $T:V \rightarrow W$ be a linear transformation. Prove that, if $\{v_1, v_2, v_3\}$ is a set of three linearly dependent vectors in $V$, then the set ...
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70 views

Polynomials in Linear Algebra

Let $n$ be a positive integer and $\mathbb{F}$ be a field. Suppose $A$ $\in$ $M_{n\times n}(\mathbb{F})$ and $P$ is and invertible matrix, such that $P \in M_{n\times n}(\mathbb{F})$. If $f$ is any ...
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A quick way to estimate eigenvector/eigenvalue of a matrix

Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could ...
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Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
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linear binary code problem

Let $\mathcal C$ be a $[n,k,d]$ linear binary code such that $\mathcal C$ has a systematic generator matrix $G=[I_k\mid A]$. (i) Prove that $u\in (\mathbb F_2)^k$ is coded by $c=(u\mid uA)\in ...
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Linear Algebra: Direct Sum

Prove that if $W_1$ is any subspace of a finite-dimensional vector space $V$, then there exists a subspace $W_2$ of $V$ such that $V = W_1 \oplus W_2$ What I have done so far is to note that ...
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2answers
18 views

Jacobian matrix for $f(h)=hh^Th$, where $h$ is an $m$ dimensional vector

I have a function $f(h)=hh^Th$, can we say $\nabla f(h)=2*hh^T + h^ThI_{m\times m}$, where $I_{m\times m}$ is an identity matrix?
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question about span and basis

I have a question from homework (I'm not sure if my solution is correct): Let $V$ be a vector space over a field $\mathbb{F}$ and let $W$ be subspace of $V$. Let $u$ and $v$ be vectors in ...
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How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$.

Suppose $\mu$ is not an eigenvalue of $A$. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
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Shown that this matrix is a representative of a group.

I have to show that this matrix (for which $ad - bc = 1$) \begin{pmatrix} a & b & 0\\ c & d & 0 \\ 0&0&1 \end{pmatrix} is a representative of a group. The same for this one: ...
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Solve this problem?

Here I have 2 lists, A and B. I am trying to find the connection items of list A got with items of list B. I know: b) all the items of A a) the 1st item of B b) number range from 0-255 A - B 0 ...
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1answer
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proving determinant of lower triangular matrix from definition

we define $$det(A) = \sum_{\sigma \in S_n} (sgn\sigma)a_{1\sigma(1)}...a_{n\sigma(n)}$$ Matrix $A = (a_{ij}) $ where $a_{ij} = 0$ for $j>i$ and I want to use it to prove the determinant of a lower ...
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counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
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Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
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Finding eigenvalues of indefinite matrix

Let V = M_2x2(R). Let T: V to V be defined by T(a,b,c,d) = (d,b,c,a). Find the eigenvalues of T and a basis B of V so that [T]_B is a diagonal matrix. I find the only eigenvalue to be 0. In this ...
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For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
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Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5 . Then, there exists a non-zero vector $v$ in $R^2$ such that (a) $||Av||$ > 2$||v||$; (b) $||Av||$ < 1/2$||v||$; ...
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Find the rank of the given matrix

Let $x_1$,$x_2$,$x_3$,$x_4$,$y_1$,$y_2$,$y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a 4 x 4 matrix A by A = $$\begin{pmatrix} x_1^2 + y_1^2 & x_1x_2 + y_1y_2 ...
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31 views

Continuity of bilinear maps

Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why? Definition of continuous ...
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Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
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Can this example have been done using row space instead of column space?

Can this example be done using row space instead of column space? I have tried but I am new so don't know if I am doing it correctly, doesn't seem right to me. I tried expressing the given vectors ...
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1answer
51 views

Prove that this matrix is not diagonalizable WITHOUT determinants

I have this matrix: $ \left( \begin{array}{cccc} 22 & 23 & 10 & -98\\ 12 & 18 & 16 & -38\\ -15 & -19 & -13 & 58 \\ 6 & 7 & 4 & -25 \end{array} \right) ...
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1answer
28 views

Change of coordinate matrix

Let $T: \mathbb{R}_2 \rightarrow \mathbb{R}_2$ be defined by $T(a,b) = (a+2b, 3a-b)$. Let $B = [(1,1),(1,0)]$ and $C = [(4,7),(4,8)]$. Find $[T]_B$ and $[T]_C$ and show that $[T]_C = Q^{-1} [T]_B Q$ ...
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Uniqueness in Matrix Multiplication

I'm sure there is an answer to this somewhere else, but I'm simply not sure how to find it or what to call it. I looked online, but couldn't find anything. The question is as follows: Let $A$ and ...
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1answer
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Help trying to identify a set and determine whether it is a subspace of $\Bbb{R}^n (n>2)$

I'm trying to figure out what this set is $\{x \mid \sum_{j=1}^{n}x_j =0\}$. Also any hints on how to show this is a subspace of $\Bbb{R}^n (n>2)$?
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Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
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1answer
21 views

Direct product norm

Given a norm on $V$ say $||*||$, what is the norm on $V \times V$? Can we induce this norm from $||*||$? Please help with understanding this.
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36 views

Determine which of the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$.

I'm having a bit of trouble showing that the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$. I know that I need to show that they are closed under addition and multiplication, ...
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1answer
31 views

linear algebra on cramer's rule

Verify the following system of linear equation in $\cos{A}$,$\cos{B}$ and $\cos{C}$ for the following triangle equation: $c\cos{B} + b\cos{C} = a$, $c\cos{A} + a\cos{C} = b$ and $b\cos{A} + a\cos{B} = ...