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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A question about product of three positive definite matrices

Assume that $A,B$ and $C$ are symmetric positive definite matrices. I guess that the eigenvalues of the matrix $D=ABC$ can be any complex numbers. Is that true?
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15 views

which of the following is an equivalence relation of the set S

which of the following is an equivalence relation of the set S I have solved all except d and need your help please
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10 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
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0answers
13 views

Jordan normal form over $\mathbb{C}$

Let there be $T:\mathbb{C}^8\rightarrow \mathbb{C}^8$ Such that $ T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} ...
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23 views

Disprove Officer's account - Velocity / Distance / Time

I am looking for a mathematical proof which definitively invalidates a false account of events, re: fail to stop - red light. If anyone is interested, thank you kindly, and please let me know if I ...
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38 views

Exercise of algebra [on hold]

Given the linear transformation $T:\mathbb{R}^4 \to P_2(\mathbb{R})$ such that: $\ker(T) = \{(x_1,x_1,x_3,x_4) \in \mathbb{R}^4 : 2x_2 - x_3 + x_4 = 0,\, 2x_1 - x_2 = 0\}$ $T(0,1,0,1) = -2x^2 + x$ ...
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A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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1answer
36 views

Three lines that intersect in a plane.

Find a condition for three lines (š‘– = 1,2,3) in a plane given by $š‘Ž_š‘– š‘„ + š‘_š‘– š‘¦ = š‘_š‘–$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
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1answer
43 views

Exercise of algebra II [on hold]

Can anybody please help me with this exercise?My exam is comming soon :S It says: 1)We define = f: Pā‚‚[R] ---> R^2x2 linear transformation whose transformation matrix in basis B and E' is: Mf(over ...
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2answers
16 views

if n is a positive integer let Z be the subset of integer in {1,…,n} which are relatively prime to n

if n is a positive integer let Z be the subset of integer in {1,...,n} which are relatively prime to n my effort to solve this question I''m confused and need help to solve this question please ...
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1answer
28 views

How does $u^Tv = p \cdot \|u\|$ follow from the projection onto line?

Before anybody asks, this is not a homework question. I just saw the formula given in Andrew Ng's Coursera course in the SVM section. For reference: the projection formula is $$ \mathrm{proj}_w(p) = ...
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1answer
18 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
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2answers
39 views

Does the set of all piecewise constant functions form a subspace of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$?

A function $f\in \mathbb{R}^\mathbb{R}$ is piecewise constant if and only if it is a constant function $x\to c$ or there exist $a_1<a_2<\cdots<a_n$ and $c_0,...,c_n$ in $\mathbb{R}$ such that ...
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0answers
20 views

Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
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3answers
21 views

Linear independence for a set of real valued continuous functions

Let $V$ be the vector space of all real valued continuous functions. Is the following set $\{\cos t, \sin t, \mathrm{e}^t\}$ linearly independent? I usually understand what and how to determine ...
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Moving object position

An object is moving with 20 points p/s. Currently the object is at position x: 30, y: 50, z: 90. The object is moving to x: 4^6, y: 4^8 z: 9. What are the coordinates after 25 minutes? This is what ...
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1answer
21 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
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11 views

Rank of product of two rectangular matrices

Given $A_{m \times n}$ matrix with rank $m$, and $B_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. We know that $$ \operatorname{rank}(AB) \leq ...
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3answers
64 views

If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$

I found an interesting conclusion when I did this simple question. Let $$f(x)=(x-1)^2(x+2)=x^3+2x^2-x-2$$ and let $x_i$ for $i=1,2,3$ be the roots of $f(x)$. Find the sum $\sum\limits_{i=1}^3x_i^2$. ...
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1answer
32 views

A question in matrix norm.

Suppose $A \in {\mathbb C^{n \times n}}$ and $\left\| A \right\| \le \varepsilon $ $v \in {\mathbb C^n}$ and ${v^*}v = 1$ Is this true that $\left\| {{v^*}Av} \right\| \le \varepsilon $?
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Determining which vectors are solutions of a given system of equations.

Determine which vectors are solutions of the system. \begin{align*} & \hphantom{+}3x-2y-5z = \hphantom{+}4 \\ & \hphantom{+}2x+4y-\hphantom{1}z = \hphantom{+\llap{$0$}}2 \\ & {-}4x-8y+9z ...
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5 views

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$.

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$. I am having difficulty in this problem please help. Here $f'(a;y)$ is the derivative of $f$ ...
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23 views

Linear Algebra Specific Solution Proof [on hold]

How do we prove that an equation follows a system of equations iff the solution of the system is a solution of the equation?
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1answer
24 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
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1answer
13 views

Algebra Linear transformations Kernel and range

if u = $(u_1,u_2,u_3) \in \Re^3 $and v $=(v_1,v_2) \in \Re^2$ be non-zero vectors, with F : $\Re^3 \to \Re^2$ by F(x) = (u.x)v. show that ker F = (span {u})$^\bot$ and that Range (F) = span {v}. ...
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Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?

Of course, it can't hold if operators are over finite-dimensional spaces, as is evident from trace considerations. Can it be true for infinite-dimensional spaces? I think not, but I don't see how we ...
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1answer
35 views

How to show T $P_2 \, \to \, P_2$ is a linear operator. Find a basis for the kernel T?

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ ā†’ $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
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1answer
27 views

A question on spectrum [duplicate]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
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Transforming a bound on vectors from unit cube back into $\mathbf R^3$

I have two vectors $\mathbf v_1$ and $\mathbf v_2$: $$\mathbf v_1 = \begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}, \mathbf v_2 = \begin{pmatrix}x_2\\y_2\\z_2\end{pmatrix}$$ If I transform the vectors ...
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The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
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Bound on maximum angle between vectors

I have two vectors $\mathbf v_1$ and $\mathbf v_2$: $$\mathbf v_1 = \begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}, \mathbf v_2 = \begin{pmatrix}x_2\\y_2\\z_2\end{pmatrix}$$ The components of these ...
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0answers
15 views

Proof that a set of vectors can be extented to a basis (but differennt from basis expansion theorem)

I have a set $S={v_i}$ of N vectors in a d ($d<N$) dimensional space (call it V), I know that those vectors span the whole space, but of course they cannot be a basis. I can expand the space to a ...
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4answers
774 views

Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on wolframs website but haven't seen any proof online as to why this is true. By orthogonal ...
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Find the kernel of T anf show that Show that R(T)=V

Let $V =\{(x,y,z) \in \mathbb R^3 : x+3y=3z\}$,and let $T :V \to \mathbb R^3$. be given by $T(x,y,z)=(x,y,z)\times(1,3,āˆ’3)$, the usual cross-product in $\mathbb R^3$. Hi i'm not sure sure about ...
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1answer
19 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
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1answer
12 views

Product of countably many 1-dimensional spaces does not have cardinality $\aleph_0$

From Bergman's "Universal Algebra: Fundamentals and Selected Topics" page 52, constructing a directly indecomposable algebra (one which does not admit a decomposition into directly indecomposable ...
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Show that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ such that $PA$ has $LU$ factorization

I am stucked at this problem: Prove by induction on $n$ that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ (a matrix obtained by rearranging the rows (or ...
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> Find the matrix A for which $[T(p(x))]_B$= for all p(x) $\in$ P2

Hey i'm quite confused with this question please link me so i can understand the theory. The question is. Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree ...
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Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
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$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ for all $a\leq x,y \leq b$. For what values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t\leq 1$, let $V_t$ be the set of all functions $f\in \mathbb{R}^\mathbb{R}$ satisfying the condition that if $a<b$ in $\mathbb{R}$ then there exists a real number ...
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If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
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Do addition and multiplication define a structure of a field? [duplicate]

I am taking an advanced linear algebra course for my Masters but never took linear in undergrad so please realize I know little to nothing about these topics. Question: Let r exist in R and 0 not ...
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The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
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1answer
52 views

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$.

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$. The multiplicative inverse is $(1,0)$. I need to show that ...
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1answer
15 views

transformation matrix between two different basis

I am working on this problem:- A rectangular coordinates $(x,y,z)$ are given in terms of new coordinates $(q_1,q_2,q_3)$ by :- $x=q_1 +q_2 \cos(\theta)$ , $y=q_2 \sin(\theta)$ and $z=q_3$. where ...
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1answer
48 views

Proof of a Vector Space

Let $F$ be a field and let $(V, +, F)$ be a vector space over $F$. If $W_1$ and $W_2$ are subspaces of $F$, prove that $W_1 - W_2 = \{v \in V | v = w_1 - w_2 \text{ for some } w_1 \in W_1, w_2 \in W_2 ...
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1answer
23 views

If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^Ī½ = Ī‘^Ī½ + Ī’^Ī½$.

I have one exercise in Linear Algebra and I would like to know if my solution is correct. If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^Ī½ = Ī‘^Ī½ + Ī’^Ī½$. My first thought is ...
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2answers
29 views

Understanding first part of dual basis proof

The textbook I'm reading attempts to proof the following: given $\left\{v_1, \ldots, v_n \right\}$ a basis for a vectorspace $V$ over $K$, there exists a basis $\left\{ \phi_1, \ldots, \phi_n ...
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1answer
25 views

Legendre transform is everywhere finite iff $ f$ grows faster than $ 2$-norm

Let $f:\mathbb{R}^n \to \mathbb{R}\cup \{ \infty \}$ be convex. Its Legendre transform is $f^* (d):=\sup_{x\in \mathbb{R}^n}(d^Tx-f(x))$ Show $f^*(d)<\infty$ $\forall d\in \mathbb{R}^n$ iff ...
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2answers
29 views

How to solve for the matrix $X$ in the following equation $AXB + X = CD$

How to solve for the matrix $X$ in the following equation $AXB + X = CD$? $A$ and $B$ are full rank symmetric matrices, and there is no structure to $CD$. $CD$ just could be $C$.