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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Find eigen value without expand and factor

I would like to compute the eigen value of the matrix: $$\begin{pmatrix}2& 0 & 4 \\ 3 & -4& 12 \\ 1& -2&5\end{pmatrix}$$ By expand the determinant of $\det(M-XId) = ...
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20 views

how to work out 3 equations simultaneously

So i was doing this linear programming question and got stuck on this part, so how do you workout simultaneously $2x + 5y = 46 $ $(2/3)x + 2y = 16 $ $(16/3)x + 4y = 64$ According to lpsolve we ...
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12 views

Proof involving vectors spaces and endomorphisms

I'm not sure how to go about this problem. Let $U$ be a finite dimensional vector space over a field. There exists some $T \in End(U)$ such that $T^2 = T$. Show that $U = U_1 + U_0$ with $T(u_1) = ...
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When told “The system reduces to one in row echelon form” what does this mean?

Does this mean that in its reduced form it is an identity matrix? Or is it describing something different?
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Prove a specific basis exists satisfying certain conditions with an endomorphism

I'm a bit confused about how I should answer this question. If someone could show me how to solve it, I would appreciate it. Let $F$ be a field and let $T \in End(F^n)$ such that $T(e_j) = e_1 + ...
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19 views

Linear transformations and matrix basis.

Hello I am currently stuck on this problem: I have no idea how to start part (i). Any help to get me started?
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29 views

Basis of $\ker T$ for $\mathcal{A}=\left\{ e^{n\cdot x}\mid n\in\mathbb{Z}\right\}$?

I am given the linear transformation $T:V\rightarrow V$ defined by $f\mapsto f''+2f'-3f$, where $V\subset\mathcal{C}^\infty$, the linear space of all real value smooth functions defined over ...
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How to prove that “If $A \in\mathcal M_{n,n}(\mathbb C)$ and $f(u,v)= @(u)&(v)$ $u,v$ an element of vector space over $C$, then $f$ is bilinear.”

How to prove that "If $A \in\mathcal M_{n,n}(\mathbb C)$ and $f(u,v)$=$@_1(u)$$@_2(v)$ $u,v$ an element of vector space over $C$, then $f$ is bilinear."
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Quick query about row spaces and basis of a matrix.

Okay so I've been given this matrix $A$ and asked to find the dimension (row $A$) and dimension (col (A)), rank($A$) then finally a basis of row($A$) and a basis of col($A$). $$A=\begin{pmatrix} 1 ...
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Vector Subspace Basis

Prove or disprove the following statement : If B = (b1, b2, b3, b4, b5 } is a basis for 525 and V is a two-dimensional subspace of R5 , then V has a basis made of just two members of B
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Reference for Affine Spaces

I recently started reading Arnold's Mathematical Methods of Classical Mechanics (Second Edition). On pg. 4 Arnold writes: Affine $n$-dimensional space $A^n$ is distinguished from $\mathbb R^n$ in ...
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Multiplicity of an eigenvalue=number of times it appears on the diagonal?

Suppose $T \in L(V)$ and $\lambda\in F$. Prove that for every basis of $V$ with respect to which T has an upper-triangular matrix, the number of times that $\lambda$ appears on the diagonal of the ...
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4 views

Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors

This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the convectors this particular basis: \begin{align} \vec{e}_0 &= x + y \tag{1} \\ ...
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Question about the K-Module $\cdot _\varphi : K[X]\times V \to V$

I managed to show several properties about the following mapping Let $K$ be a field, $V$ a finite dimensional $K$-Vectorspace and $\varphi \in \text{End}_K(V)$. $$ \cdot_\varphi : ...
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1answer
20 views

Radon space and Polish space.

This should be simple but could not understand. This refers to the link: http://en.wikipedia.org/wiki/Polish_space where it mentions that: Lusin spaces, Suslin spaces, and Radon spaces are ...
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1answer
15 views

How can it be shown that $\vec{w} + \vec{v}$ is either an eigenvector of a symmetric matrix or equal to the zero-vector?

How can it be shown that $\vec{w} + \vec{v}$ is either an eigenvector of H or equal to the zero-vector? I'm not sure how to approach this. Here are the details given: I is a 3x3 identity matrix. P ...
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32 views

how to show a map is linear

How do I show that the map $T:\Bbb R^3 \to \Bbb R$ defined by $$ T\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=x_1-2x_2+x_3 $$ is linear? I know that I have to show that it is closed under addition ...
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1answer
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Finding Linear Independence in $P_3$

We are asked to deteremine whether the vectors are linearly independent: $ x^2 + 1, x + 1, x^2 + x $ in $P_3$ I began the problem as follows: Let $ S $ = { $x^2 + 1, x + 1, x^2 + x$ } in $P_3$ ...
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2answers
79 views

If I know $AB$, how can I calculate $BA$?

Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2\times3}(\mathbb{R})$ be matrices satisfying $AB =\begin{bmatrix} 8 &2 &−2\\ 2 &5 &4\\ −2 &4& 5 \end{bmatrix}$. ...
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1answer
11 views

Componentwise product of two matrices and the component wise product of their symmetric parts

Given two matrices $A$ and $B$, we have their Symmetric parts $\hat{A}=1/2(A+A^T)$ and $\hat{B}=1/2(B+B^T)$. Is there a simple relation between their componentwise product? i.e. Are ...
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25 views

What does it mean if the standard Hermitian form of complex two vectors is purely imaginary?

If $v,w \in \mathbb{C}^n$, what does it mean geometrically for $\langle v , w \rangle$ to be purely imaginary?
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1answer
45 views

What does the notation $A:B$ means for matrices $A$ and $B$?

What does the notation $A:B$ means for matrices $A$ and $B$? I saw this is an equation derived from the Navier-Stokes equation. For example, \begin{equation*} \int_{\Omega}\nabla(u):\nabla(v)dx ...
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28 views

To find jordan canonical form

which of the following matrices have Jordan canonical form of equal to the 3*3 matrix $$ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ a)$ ...
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8 views

How to find parameters from logistic equation

I have an function and assume that that is convex function. I want to use gradient decent to find parameters in that equation. Could you suggest to me the way to do it. Thanks. This is my function ...
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jordan canonical form… [on hold]

I don't understand how to apply jordan canonical form of a matrix. Please Guide me with some examples or some reference regarding it. Thanks...
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76 views

I have the Eigenvalues, how do I get Eigenvectors?

My matrix is \begin{array}{ccc} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{array} Through the rule of Sarrus, I know (approximately) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ ...
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1answer
16 views

Distinction between algebra homomorphisms and $A$-module homomorphisms

I am getting quite confused about the distinction between algebra homomorphisms and $A$-module homomorphisms, where $A$ is an algebra. If $A=\mathbb CG$, the group algebra, then I have a result in my ...
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1answer
19 views

Existence of an Isometry

Suppose $T_1,T_2$ are normal operators on $L(F^3)$ and both operators have $2,5,7$ as eignevalues. Prove that there exists an isometry $S$ such that $T_1=S^*T_2S$. I know that since they have the ...
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Equal number of poles and zeros for square transfer matrix

In page 10. of the docuement below (MIT Courseware on control) it is stated that since the transfer function is square, there is an equal number of poles and zeros. Does this hold and if so under ...
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Cramer's Rule Proof Question

I have read the following proof on Wikipedia How does $X_1$ columns are $A^{-1}b,A^{-1}v_2,...,A^{-1}v_{n'}$ are they to columns augmented? or are they matrix multiplication ?
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Algebra:Linear Equations in one Variable

I study in 9th standard. I'm dealing with a word problem. The tens digit of a two digit number exceeds its unit digit by $4$. If the tens digit and units digit are in the ratio $3:1$, find the ...
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What's the word for a number which is used to scale down a value?

I'm a programmer and I'm creating an API in which there is a parameter the user can pass in which scales down a value. So for example: ...
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2answers
34 views

Let $A$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
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Let $V$ be the vector space over $R$ composed of all polynomials in **R[X]** having degree less than 3 [on hold]

can someone help me with this problem please. Let $V$ be the vector space over $R$ composed of all polynomials in R[X] having degree less than 3 and let $W$ be the vector space over $R$ composed of ...
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1answer
12 views

Projection out of orthogonal matrices

Let A,B be orthogonal matrices of order $n \geq 2 $. $\det A = 1, \det B = -1$. There exist $a \in [0,1]$ such that $aA + (1-a)B$ is projection. I know that the claim above is false. I ...
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1answer
37 views

Isn't it a subspace?

I have a problem with the concept of subspace. Determine that following sets are subspaces of $R^2$. (1) $W = \{(a,b+1)|a,b \in R\}$ (2) $V = \{(a+2b,b+1)|a,b \in R\}$ I know $W$ ...
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1answer
9 views

Variety of maximal isotropic subspaces

Suppose that $V$ is a complex vector space of even dimension $2n$. Let $Q:V \times V \rightarrow \mathbb{C}$ a bilinear, non degenerate, simmetric bilinear form over the field of complex number. Set ...
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1answer
14 views

Second Derivative Test and Hessian for $f(x,y) = x^2 + y^2$.

My task was to find the critical points of the function $f(x,y) = x^2+y^2$, to then compute the Hessian, and to use the second derivative test to determine whether the critical points are local maxima ...
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1answer
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interpolating and difference table, an old mid exam?!

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, how many fraction was used? ...
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3answers
57 views

Equivalence of $\|x\|_1\|x\|_{\infty}$ and $\|x\|_2^2$

Let $x$ be any complex $n$-vector and let $\|\cdot\|_p$ denote the usual $p$-norm. It is easy to show that $\|x\|_2^2\leq\|x\|_1\|x\|_{\infty}$ (Hölder's inequality). What I am rather interested in is ...
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Matrix Properties Reference

A lot of proofs I come across (working on stability of numerical methods) apply some property of a particular type of matrix. This includes, for example, the fact that the $L_{2}$ norm of a normal ...
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Matrix Change of Basis

guys. I'm not entirely sure how I'm not getting the right answer for this question. I'll try to explain what I've tried so far. I need to computer MB1->B2 and MB2->B1 B1 = {(0,0,1),(1,0,0),(0,1,0)} ...
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Tell me whether this Unknown Operation Exists.

I need to know whether the below unknown operation, denoted by $\boxplus$ exists. If $v_1=a \boxplus X$ and $v_2=b \boxplus X$, where $X$ is an identical value in both $v_1$ and $v_2$: equation (1): ...
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Solving LES containg spherical coordinates

i have a three-dimensional parametric equation of a line, where the directional vector is normalized and converted to spherical coordinates to calculate a angle offset. It looks like this: ...
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1answer
12 views

Orthogonal set proof?

Isn't this just the definition of an orthogonal set? What needs to be done to actually prove this?
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1answer
50 views

Prove that A(AB-BA) = (AB-BA)A implies AB-BA is nilpotent.

Let A and B be $n \times n$ complex matrices such that $A(AB-BA) = (AB-BA)A$ a) Show that for every positive integer $k$, the matrix $(AB-BA)^k$ is of the form $AC-CA$, where $C$ is an $n \times n$ ...
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1answer
29 views

Equation for adjoint transformation and proof.

I am really lost on this one. Any help would be appreciated. I'm very confused.
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8answers
857 views

To find eigenvalues

find the eigenvalues of the $6\times 6$ matrix $$\left[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 ...
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24 views

Transformation self adjoint proof

Let $T$ be a linear operator on an inner product space $V$. Let $U_1 = T+T^*$ and $U_2 = TT^*$. Show that $U_1$, $U_2$ are both self-adjoint. I understand these just as innate properties. I don't ...
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Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct prime positive integers, or show that no such matrix can exist.

I know that the matrix exist because the entries are primes but I don´t know how to explain, i need some help. Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct ...