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 ...

learn more… | top users | synonyms

0
votes
0answers
4 views

Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
1
vote
2answers
9 views

How do you show that f(z)=z conjugate isn't linear?

let x1= a+bi let x2= c+di let k=scalar f(x1,x2)=f(x1) + f(x2) f(a+bi + c + di)=(a+c)-(b+d)i f(a+bi)+f(c+di)=(a+c) - (b+d)i f(kx1)=kf(x1) f(k(a+bi))= k(a-bi) kf(x1)=k(a-bi) Looks linear to me, ...
0
votes
0answers
3 views

Minimum of restricted linear combinations.

Let $\{N_0, ... , N_m\}$ be a set of natural numbers, then the minimum $(\geq 1)$ of all their linear combinations is their GCD. Is there a way to calculate that minimum if some $N$s can only be ...
0
votes
0answers
5 views

Given the tetrahedron $OABC$, find a condition on $a OA+ b OB + c OC$ such that this is always inside $ABC$.

I did the following: Taking the tetrahedron $OABC$, one can decompose it in: $OA,OB,OC, AB,BC$. And then, writing: $$x(BC-AB)+AB\quad x\in[0,1]$$ We obtain all the points in the line segment from ...
-1
votes
1answer
30 views

I want to appeal this problem from an exam in Linear Algebra I, do you think its appealable?

I have the follow question : Let $U_1, U_2, W$ are linear spans of linear space $V$ while V is finite. Proof: If $$U_2 \cap W \neq \{0\}$$ $$U_1\cap W\neq \{0\}$$ $$U_1 \cap U_2=\{0\}$$ Then $dimW ...
0
votes
1answer
22 views

Complex conjugate of $z$ as a different variable

Can a complex conjugate be represented by a different letter than $z$? As in: Let $y$ be a complex number satisfying $|y|<1$. Find the set of all complex numbers $z$ satisfying ...
0
votes
0answers
15 views

What is a change of basis and how do i find it?

W is a four dimensional vector space over a field F with basis S = (v1, v2, v3, v4). B is also a basis of W such that. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...
1
vote
1answer
23 views

Where do the variables of a quadratic form live?

Consider a quadratic form $Q = a_{ij}x_{i}x_{j}$, where the summation from 1 to $n$, the number of independent variables is implied on $i$ and $j$. By this definition $a_{ij}$ is not symmetric, but ...
0
votes
0answers
10 views

Is the following optimization problem a way to find SVD?

Let $B$ a matrix of size $n \times m$. I want to show that the SVD, top $k$ singular vectors, can be found by solving: $$\max_{\displaystyle{\begin{array}{c}||u_i|| = 1, i \in \{ 1,\ldots, k \},\\ ...
1
vote
0answers
28 views

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?
0
votes
1answer
17 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
0
votes
1answer
23 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 $2$ ...
2
votes
0answers
19 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} ...
-2
votes
0answers
24 views

Disprove Officer's account - Velocity / Distance / Time [on hold]

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 ...
-5
votes
0answers
39 views

Exercise of algebra [on hold]

Given the linear transformation $T:\mathbb{R}^4 \to P_2(\mathbb{R})$ such that: $\ker(T) = \{(x_1,x_1,x_3,x_4) \in \mathbb{R}^4 : 2x_2 - x_3 + x_4 = 0,\, 2x_1 - x_2 = 0\}$ $T(0,1,0,1) = -2x^2 + x$ ...
-2
votes
0answers
15 views

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 ...
-1
votes
1answer
41 views

Three lines that intersect in a plane.

Find a condition for three lines (𝑖 = 1,2,3) in a plane given by $𝑎_𝑖 𝑥 + 𝑏_𝑖 𝑦 = 𝑐_𝑖$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
-5
votes
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: P₂[R] ---> R^2x2 linear transformation whose transformation matrix in basis B and E' is: Mf(over ...
0
votes
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 ...
2
votes
1answer
34 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) = ...
5
votes
1answer
19 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 ...
3
votes
2answers
44 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 ...
0
votes
0answers
21 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, ...
2
votes
3answers
23 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 ...
-1
votes
0answers
17 views

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 ...
0
votes
1answer
23 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 ...
0
votes
0answers
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 ...
6
votes
3answers
75 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 $\sum\limits_{i=1}^3x_i^2$. ...
2
votes
1answer
33 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 $?
1
vote
0answers
9 views

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 ...
0
votes
0answers
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 $f$ ...
-1
votes
0answers
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?
0
votes
1answer
30 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 ...
0
votes
1answer
15 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}. ...
5
votes
3answers
35 views

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 ...
1
vote
1answer
36 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 : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
0
votes
1answer
28 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) = ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
21 views

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 ...
2
votes
0answers
33 views

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 ...
1
vote
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 ...
3
votes
4answers
857 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 ...
0
votes
1answer
20 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 ...
0
votes
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 ...
0
votes
0answers
23 views

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 ...
0
votes
0answers
15 views

> 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 ...
0
votes
0answers
17 views

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 ...
0
votes
0answers
25 views

$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ 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 ...
0
votes
0answers
24 views

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 ...
0
votes
0answers
29 views

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 ...