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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Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ \mathrm{...
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3answers
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Prove that matrices $\tiny\begin{pmatrix} 2&-1 \\ 0&2 \\ \end{pmatrix},\begin{pmatrix} 2& 0 \\ 1&2 \\ \end{pmatrix} $ are similar. Error in my method?

Show that the matrix $$\begin{pmatrix} 2&-1 \\ 0&2 \\ \end{pmatrix} $$ is similar to a triangular matrix of the form $$ \begin{pmatrix} \lambda& 0 \\ 1&\lambda \\ \end{pmatrix} $$ ...
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1answer
52 views

Action by Orthogonal Matrices is Transitive

Let $A$ and $B$ be $n \times k$ real matrices with orthonormal columns, where $1 \leq k \leq n$. Suppose $\text{col}(A) = \text{col}(B)$ and $A^TA = B^TB = I_k$. I want to show that there is a $k \...
0
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0answers
32 views

Find linear transformation without factoring its characteristic polynomial (if given)

Find a linear transformation $\text{T}: \mathbb{R^5}\to \mathbb{R^5}$ whose characteristic polynomial is $p(t)=-(t^5-3t^3+t)$ without factoring $p(t)$. Does it have anything to do with canonical ...
2
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2answers
61 views

Prove order N symmetric matrix is negative definite, and the second order derivative of matrix

I have three questions to ask, hope to get some help from you (1) Is there any effective way to determine the negative definiteness of a N-order matrix $\textbf{M}$? I think it's impossible to solve ...
0
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0answers
81 views

Estimating the value of a stock

I need a way to know the value of my stock. Let $(x_1, \dots, x_n)$ be the quantity of the products $1$ to $n$ I have in stock, such that, for example, if I have $8$ units of the product $2$, $x_2 = 8$...
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1answer
57 views

Proof that gradient of $\det(A)$ with respect to $A$ is $\det(A) A^{-1}$ [closed]

How to prove that $\dfrac{\partial |A|}{\partial A} = |A|A^{-1}$ where $|A|$ is $\det(A)$ and $A$ is symmetric matrix?
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0answers
42 views

Alternative proof of existence Jordan normal form

Consider the next theorem: Let be $E$ is an $n$-dimensional vector space over $\mathbb R$ and $\alpha$ a 2-vector. Then there is a basis $\sigma_1,\sigma_2,\ldots,\sigma_n$ such that $$\...
4
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1answer
51 views

Computing bases for direct, wedge, tensor products, etc., of given vector spaces

I am filled with all kinds of vector space and I want to make sure I understand the basis for each kind of vector space. Suppose $\{v_i\}_{i=1}^n$ is the basis for vector space $V$, $\{w_j\}_{j=1}^m$ ...
1
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1answer
183 views

vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
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1answer
36 views

Integer-valued polynomial question

Let us have an $f(x)$ Integer-valued polynomial, which gains the value $1$ in $4$ different places. Prove, that in that case, it can't gain the value $-1$ on integer places. I tried with $f(x)-1=0$, ...
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3answers
87 views

Proving that matrices $B$ produced by transposing two rows of $I$ satisfy $B=B^{-1}$?

For example, I have a $3 \times 3$ identity matrix. If I exchange rows $2$ and $3$, then I get $$ B = \pmatrix{1&0&0\\0&0&1\\0&1&0}. $$ In this case it can be checked that $B=...
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3answers
38 views

How to prove this statement with A,B matrices? [duplicate]

Let us have two matrices: $A,B$ which are both $n$ x $n$ type, and $AB+A+B=0$ stands. Prove, that in this case, $AB = BA$. I tried to do such matrices, and the statement stands, but how should I do a ...
2
votes
2answers
311 views

reciprocal vectors

I don't understand some of the terminology in this question. I googled reciprocal vectors and got an article on reciprocal lattices, but I'm not sure if that is what they are talking about in this ...
2
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1answer
75 views

linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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1answer
34 views

Some detail needed in Positive Definite Matrix

First all of all, I am sorry I have put a page of my text book. Somehow I need some help to understand some paragraphs in the page. If you can explain, please let me have some explanations to those in ...
2
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3answers
48 views

Linear Algebra Question for Repeated Eigenvalues

Can somebody explain what the sentence in the red circle means?
2
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1answer
68 views

Solutions to $u \circ v - v \circ u = \mathrm{Id}$

Let $(E,\Vert \cdot \Vert)$ be an infinite-dimensional normed vector space and $\mathcal{L}_{c}(E)$ denote the ring of continuous endomorphisms of $E$. I would like to determine whether the equation ...
2
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1answer
46 views

How does the following line of code Simplify to …?

How does Simplify to I don't get the middle part... Please give me a explanation.
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1answer
32 views

How do I see that for a matrix $A \in \text {Mat}_{m \times n}(\mathbb R)$ the following holds: $\lim_{x \rightarrow x_0} A(x-x_0) = 0?$

How do I see that for a matrix $A \in \text {Mat}_{m \times n}(\mathbb R)$ the following holds: $\lim_{x \rightarrow x_0} A(x-x_0) = 0?$ I see that $\lim_{x \rightarrow x_0} A(x)-A(x_0) = (\lim_{x \...
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1answer
51 views

Is the transformation $T: (r, \theta) \to (r, \theta + \phi)$ linear? Here $\phi$ is a given angle

Let $T$ rotate every point through the same angle $\phi$ about the origin, $i.e.$ $T: (r, \theta) \to (r, \theta + \phi)$ where $\phi$ is given. If in addition that $T(O) = O,$ namely, if $T$ maps the ...
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0answers
59 views

Real and imaginary part of an Eigenvector.

Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) generated by a band limited signal on a set of sensors. Is there a relation ...
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0answers
65 views

Induction proof: $\det(M) = \prod_{1 \le j \le n} (x_j - x_i)$

Following problem: Let $\mathbb{K}$ be a Field and $M = \begin{pmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & \vdots & & \vdots \\ 1 & x_n & \...
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1answer
37 views

Show that ||T(M)|| $ \le$ ||M|| for all M

This is the last part of a very long question. $$ M = \begin{pmatrix}a&b\\c&d\end{pmatrix}\hspace{3pc} \text{and} \hspace{3pc} T(M) = \frac{1}{2}\begin{pmatrix}a +id&b+ic\\c-ib&d-ia\...
3
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1answer
62 views

Prove that $\operatorname{null}T^m=\operatorname{null}T^{m+1}$ if and only if $\operatorname{range}T^m=\operatorname{range}T^{m+1}$.

$\newcommand{\range}{\operatorname{range}}\newcommand{\null}{\operatorname{null}}$Problem: Suppose $T\in\mathcal{L}(V)$ and $m$ is a nonnegative integer. Prove that $\null T^m=\null T^{m+1}$ if and ...
2
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3answers
41 views

Interpreting $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $ as a scalar or matrix multiplication

As part of another problem I am working on, I have the following product to work out. $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \cdot h $ where $h$ is a scalar. My question is, if I commute ...
0
votes
2answers
452 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - y)$...
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1answer
54 views

Finding shortest distance on earth

This is a math project for my linear algebra class. I have been having troubles figuring out if my answers are correct. I am using the dot product to figure out the great circle distance between two ...
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1answer
26 views

Terminology with linear transformation

I am working on a problem that asks me to "Write C for the matrix whose ij entry is $(1/2)^{ij}$" given that $M$ is the vector space of all $n x n$ matrices and $l$ is a linear transformation on $M$. ...
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3answers
119 views

What disqualifies an expression from being linear?

I'm taking some advanced math classes at my high-school, and I have some questions. A helpful answer would include a definition. Examples of a concept or definition given (no matter how simple; ...
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1answer
109 views

Understanding Jordan Canonical Form.

Two questions: How does the nilpotent index $k$ of a linear transformation L on a vector space of dimension $n$ relate to possible Jordan Canonical Forms? My understanding is that a Jordan block ...
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2answers
372 views

The matrix P is the transition matrix from what basis B to the basis B'

The Matrix \begin{equation} P = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 1 \end{bmatrix} \end{equation} is the transition matrix from what basis B to the basis B' = {(...
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1answer
70 views

Gram-Schmidt Process derivation question

I have a problem with this derivation I hope you can help me with: so we have constructed $w_{m+1}$ such that $g(w_{m+1},w_{m+1}) = 1$ and 0 otherwise, and we have show that $w_{m+1}$ is well defined,...
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2answers
52 views

Proof about linear functionals without reference to a basis.

Recall the following: Let $V$ be a vector space, and let $v$ be in $V$. If $\lambda(v) = 0$ for all $\lambda \in V^*$, then $v = 0$. Does anyone know of a proof that makes no reference to a basis? ...
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2answers
76 views

How do I Solve this Seemingly Simple Set of Four Equations with Four Unknowns?

I have what looks like a set of simple simultaneous equations: 4 equations with 4 unknowns. The numbers are really simple, and in fact I already know the answer, but I cannot figure out how to work ...
3
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0answers
56 views

Relation between $y=Ax_1$ and $y=WAx_2$

I have a question. Is there any relation between the following linear equations? $$y=Ax_1 \ \ \text{ and} \ \ y=WAx_2$$ W is diagonal square invertable matrix, A is an mxn matrix with $n>m$. I ...
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0answers
16 views

What happens when you add linear operators or take one linear operator inside another?

Let $A$ and $B$ be linear operators on $\mathbb R^2$. $A$ is the projection operator on the x-axis and $B$ is the the counterclockwise rotation by angle $\frac{\pi}{6}$. Find matrices of linear ...
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1answer
43 views

Matrix endomorphism rank

Let $(A, B) \in M_n(\mathbb{C})^2, u \in L(M_n(\mathbb{C})) \text{ > s.t. } \forall M \in M_n(\mathbb{C}), \text{ } u(M)=AMB$. 1) Find $\det(u)$ 2) Find $\mathrm{rk}(u)$ 1) $\det(u) = \...
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1answer
59 views

I want to know why $rank \: M=1$

Let $A$ be a non-zero and non-invertible $n\times n$ matrix. Suppose that $M$ is a non-zero $n \times n$ matrix such that $MA=AM=0$ . I want to know why $rank \: M=1$.
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0answers
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Find Jordan canonical form with Kronecker product of JCF

Let $f: K^3\to K^3$ be a map in Jordan canonical form having a matrix $$\begin{pmatrix} 2 & 1 & 0\\ 0 &2&0\\ 0&0&2\\ \end{pmatrix}$$ Find the JCF of the map $f\otimes f$. So,...
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0answers
71 views

a special matrix inverse

Let $A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an ...
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0answers
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Finding proper matrix?

I would like to find a "mechanic" way in order to solve such questions. Find a matrix $A \in \mathbb{R}^{3\times3}$ corresponding to the following: $ A\cdot A=$ \begin{pmatrix} 1 & 0 &2 \\ 0 ...
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1answer
36 views

Proving using linear transformation

This is the question : Prove or disprove : If V is a linear space and T: V → V is a linear transformation such that T^2 = 0 (the zero transformation), than Im(T) ⊆ Ker(T) I thought about it a lot ...
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1answer
56 views

The dimension of the SU(2) matrix group

Let's take the matrix $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Consider its transpose $R^\dagger = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix}$. Then $RR^\dagger =...
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1answer
46 views

Eigen values of the operator $T : V \rightarrow V : T(f(t)) = t f~'(t)$

Let $V$ be the linear space of all real functions differentiable on $(0,1)$. If $f \in V,$ define $q = T(f(t))$ to mean that $q(t) = tf~'(t) ~\forall ~t \in (0,1)$ Prove that every real $\lambda$ is ...
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2answers
38 views

Is rotation in $\mathbb{R}^d$ unique?

Let $\boldsymbol{u} \in \mathbb{R}^d$ such that $||\boldsymbol{u}||_2 = 1$ be a directional vector. Let $Q_{\boldsymbol{u}} \in \mathbb{R}^{d \times d}$ be an orthogonal matrix such that $Q_{\...
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1answer
106 views

True/False Question About Whether A Linear Transformation is onto.

We are given that V, W are vector spaces. We are told that $v$1, $v$2 are distinct vectors in V and that $w$1, $w$2 are distinct vectors in W. The first True/False question states: 1) There is a ...
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0answers
120 views

Does gaussian elimination always work?

If so, why don't we use that to get from any square matrix to a triangular matrix - from which can be deduced eigenvalues, determinant (product of eigenvalues) and diagonal matrix (since the diagonal ...
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5answers
1k views

Is there anything special about this matrix?

I've just encountered a matrix which seems to display nothing special to me: $$B=\begin{pmatrix}1&4&2\\0 &-3 &-2\\ 0 &4 &3 \end{pmatrix}$$ But further observation reveals ...
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0answers
75 views

Computing the matrix representation of the quadratic form $A \mapsto \text{tr}(A^2)$

Define the quadratic form $Q:\mathbb{R}^{2\times 2}\to\mathbb{R}$ by $$Q(A) = \text{tr}(A^2).$$ What is the matrix representation of this bilinear form with respect to the standard basis of $\mathbb{R}...