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 the adjoint of the Linear OperatorT

Find the adjoint of the Linear Operator T:R3-R3 defined BY T(x,y,z) = (x+2y,3x-4z,y)
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1answer
7 views

The projection $EF=E$ imply $M_2\subset M_1$?

Suppose $F$ is a projection on $M_1$ along $N_1$, $E$ is a projection on $M_2$ along $N_2$, if $EF=E$, does that imply that $M_2\subset M_1$?
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Is the basis for $\mathbb{R}^n$ where the $\mathfrak{so}(n)$ Cartan elements are diagonal, necessarily complex?

This is a follow-up to this and this question. The elements of $\mathfrak{so}(n)$ are antisymmetric in the standard basis $(1, 0, 0), (0, 1, 0), (0, 0, 1)$. This means that we have no diagonal ...
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1answer
22 views

$U^TA_1V$ is a rank-one matrix?

To give a little bit of context, the question I am asking is related to SVD decomposition. More specifically, we are trying to prove that the best rank one approximation for $A_1$ is $\sigma_1 u_{1} ...
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17 views

Can we express any matrix as an outer product expansion?

Suppose $XY$ is an $m $ by $n$ matrix, where $X$ is a $m$ by $k$ matrix and $Y$ is a $k$ by $n$ matrix. $y_i$ are the columns of $Y$ and $x_i$ are the columns of $X$. How do we know that ...
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1answer
38 views

Quadratic forms and midpoints

The midpoint of the vectors $u$ and $v$ is $w=\frac{u+v}{2}$. In euclidean geometry, an alternative characteristic of midpoints is $|v-w|=|u-w|=\frac{1}{2}|u-v|$. I wonder if this generalizes to ...
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27 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
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40 views

What do accountant's learn? [on hold]

Since a high school student can find compound interest, calculate stock yields, etc. what does an accountant learn in college? Is it just busy work?
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2answers
24 views

Closed under scalar multiplication [on hold]

The subset of $\mathbb{R}^2$: $\{ (x,y)| y=\frac{7}{2}x\}$ is a subspace of $\mathbb{R}$. How can I prove that the subspace is nontrival ?
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40 views

Diagonalization and Commuting Matrices

Attempt: I have shown part (ii) and I have found the eigenvalues and eigenvectors for A, B respectively and shown they can be diagonalised. I need help with (iii) and (iv), for (iii) I can't show ...
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2answers
28 views

How to find $f$ for a symmetric bilinear form?

Let's say we have the symmetric matrix:$$A = \left(\begin{array}{cc} 1&2 \\ 2&0 \end{array}\right)$$ How do I find the symmetric bilinear form of this $A$?
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What is the fastest, most correct way to solve this simultaneous of two linears?

\begin{eqnarray*} (x+2)/5-((y+2)/4) &=& 2-(x/3) \\ (x+5)/4+((x-y)/5) &=& y+5 \end{eqnarray*} What is the fastest, most correct way to solve this simultaneous of two linears?
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6answers
78 views

Find $x$ and $y$ - Why is there no answer?

I need to find $x$ and $y$ from the following equations: \begin{eqnarray*} 7x-3y &=& 8 \\ 14x-6y &=& 21 \end{eqnarray*} I my book it says there's "no answer". Can someone explain to ...
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2answers
41 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...
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1answer
14 views

minimal polynomial and invariant subspace

I'm trying to solve the following problem: Let $T$ be a linear transformation on a finite dimensional vector space $W$. Suppose the minimal polynomial of $T$ is $p=g_1g_2$, where $g_1, g_2$ are ...
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1answer
35 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and (A+B) UPDATE: Assume $\bf (A+B)$ is invertible. and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf ...
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3answers
63 views

Does every invertible complex matrix have an eigenvector? [on hold]

Over $\mathbb{C}$ does every invertible matrix have at least one non-zero eigenvalue and an eigenvector? I'm generally confused about eigenvectors and eigenvalues. I understand that eigenvectors are ...
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1answer
20 views

Eigenvalues of Certain Symmetric Block Matrix

What can we say about the relation between the eigenvalues of the following block matrix with identity diagonal blocks, and the singular values of the off-diagonal blocks: \begin{equation} ...
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2answers
32 views

All the cases for Image and Kernel

here alpha is a real variable, and I need to find the kernel and image for all values for alpha. Attempt: I can't seem to figure out all the cases which I need to evaluate, as the last two columns ...
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3answers
46 views

What is the definition of a $\mathbb{F}_2$-linear function?

To clarify, the function is $f:\mathbb{F}_2^m\rightarrow \mathbb{F}_2$. So, does it mean linear in each variable, or perhaps that each monomial is of degree $\leq1$? I know that sometimes terms have ...
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1answer
17 views

Interlacing Theorem on Singular Values

Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the ...
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0answers
21 views

Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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1answer
19 views

Why $(U+iI_n)^2 = (U+iI_n)(U^*-iI_n)$?

Let $U$, a unitary operator and let $U+iI_n$, self-conjugate operator. Why is it true that: $$(U+iI_n)^2 = (U+iI_n)(U^*-iI_n)$$ We can evaluate both sides of the equation to get: $$(U+iI_n)^2 = U^2 ...
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2answers
36 views

If $UT=TU $, why is the range of $U $ invariant under $T $?

My Linear Algebra book says the following: Let $V$ be a vector space and $T$ be a transformation, which commutes with another transformation $U$. Then the kernel and range of $U$ are invariant ...
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1answer
90 views

Why we wonder to know all derivations of an algebra?

It is well-known that the space of all derivations of algebra of smooth functions on a manifold is its space of sections (vector fields on underlying manifold). But, I wonder to know why finding the ...
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1answer
48 views

$A,B$ are two real positive matrices then $\det (A+B) > \max(\det A , \det B)$

Let $A,B$ two square-real-positive matrices. Prove that $\det (A+B) > \max(\det A , \det B)$ So I found this solution: http://math.stackexchange.com/a/41478/160028 Basically, if $A=I_n$ and $B$ ...
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4answers
84 views

Is there always a matrix $X$ such that $X^2=A$?

Is it true that for every $A\in M_{2\times 2} (\mathbb{C})$ there's an $X\in M_{2\times 2} (\mathbb{C})$ such that $X^2=A$? For the matter of fact, I don't have a clue, other than evaluating the ...
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1answer
25 views

Proving that same solution set implies row equivalence

The question I am trying to solve is for a much simpler case: Suppose $R$ and $R'$ are $2\times3$ row-reduced echelon matrices and that the system $RX=0$ and $R'X=0$ have exactly the same ...
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2answers
42 views

Are those matrices congruent?

$$A = \left(\begin{array}{cccc} 1&0 \\ 0&-1 \end{array}\right), B=\left(\begin{array}{cccc} 1&0 \\ 0&2 \end{array}\right), C = \left(\begin{array}{cccc} 1&0 \\ 0&4 ...
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4answers
52 views

Vector Subspace

I have a question regarding vector subspaces: show $U=\{A\in M_{22} \mid A^2=A\}$ is not a subspace of $M_{22}$. I have said: let $A={(a_1, a_2, a_3, a_4) = \begin{pmatrix} a_1 & a_2 \\ a_3 & ...
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3answers
23 views

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$ I'm unsure how to proceed with proving this. Basically my idea is as ...
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1answer
19 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
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0answers
9 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
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2answers
38 views

Badly formed question? $\|x\|=1=\|y\|$ and $\|x+y\|=\|x\|+\|y\|$, there is a line segment in the unit sphere

Show that if a normed linear space $X$ contains linearly independent vectors $x$ and $y$ such that $\|x\|=1$ and $\|y\|=1$ with $\|x+y\|=\|x\|+\|y\|$, then there is a line segment contained in the ...
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1answer
40 views

What is $\frac{\partial x^TA^{-1}y}{\partial A}$?

I had trouble proving the following: If $ A\in \mathbb R^{n\times n}$ and $A$ is nonsingular, $x \in \mathbb R^{n\times 1}$, $y \in \mathbb R^{n\times 1}$, then $ \dfrac{\partial ...
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Does solution to particular system of O.D.E. with boundary conditions exist and unique?

I am dealing with the following system of $2$nd order differential equations: $$\ddot{y_k}+\frac{\ddot{y}_{N,k}}{2}+a\dot{y}_{N,k} = \gamma_k^2 y_k$$ where $y_{N,k} = \sum_{i=1,i\not=k}^{N} y_i ...
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3answers
122 views

Can we prove that matrix multiplication by its inverse is commutative? [duplicate]

We know that $AA^{-1} = I$ and $A^{-1}A = I$, but is there a proof for the commutative property here? Or is this just the definition of invertibility?
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11 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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1answer
56 views

Direct Sum Proof

I am reading Axler's Linear Algebra Done Right Book and I am on the part about the direct sum. He gives the following proposition: When he assumes that $a$ and $b$ hold to prove that the proof gives ...
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2answers
38 views

The $\exp \circ \log$ function acts as the identity on unipotent matrices.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 9 of chapter 2 using the provided hint. In chapter 2, Hall defines the ...
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Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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1answer
29 views

Is linear independence preserved through column transformations?

Let's say you have a $m\times n$ matrix $A$, and the $n$ column vectors are linearly independent. And let's say you have a transformation $T$. You perform the transformation on each column of $A$, ...
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2answers
36 views

linear algebra and solving for one solution.. [on hold]

Consider the system of linear equations $(\lambda -3)x+ y=0$ $\hspace{0.3cm} x+(\lambda -3)y=0$ Determine the value(s) of \lambda such the system has: 1) infinitely many solutions; 2) exactly one ...
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4answers
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linear algebra and solving has infinitely many solutions. [on hold]

Determine the value(s) of $k$ such that the system of equations $$4x+ky=6$$ $$kx+y=-3$$ has infinitely many solutions.
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1answer
24 views

Prove for symmetric real matrix $M$,$z^TMz>0$ for real vector $z$ for real implies it's true in complex

For symmetric real matrix $M$, $z^TMz>0$ for real vector $z$, how to prove that it is also positive definite for $z\in C$?
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1answer
27 views

linear algebra find max and symmetric matrices

I am working through the following problems and have gotten stuck. I can do (1) and (2) for both groups but am not sure how to go about doing the other questions(find max, and the two questions for ...
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1answer
69 views

Finding $a^{2014} + b^{2014} + c^{2014}$ given some conditions on $a,b,c$.

I came across this problem: "Let $a$, $b$, $c$ be nonzero real numbers that satisfy the conditions : $$a + b + c = 9,\\\mathrm{and}~ab + bc + ca = 27 $$ Calculate $$a^{2014} + b^{2014} + ...
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1answer
36 views

Two conflicting answers: Problem in linear algebra involving quotient spaces and T-invariant subspaces

I was presented this scary looking problem in my linear algebra class involving quotient spaces: I am given finite dimensional vector space V over the complex numbers C and linear operator $ T:V ...
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1answer
26 views

A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
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1answer
38 views

Generalization of Lines and Planes

Let $a_1,a_2,\dots,a_n$ be constants that are not all zero. An equation defines a line if and only if it can be written: $a_1x_1+a_2x_2+a_3=0$ An equation defines a plane if and only if it can be ...