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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43 views

Confused about matrix and vector operations, properties.

I am currently studying topics in Machine Learning and came across a solution I do not fully understand. The problem #4a, the statement and solution can be found here: http://cs229.stanford.edu/...
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2answers
188 views

Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
2
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1answer
30 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
2
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2answers
44 views

Is a subset of an inner product space also an inner product space?

My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I ...
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0answers
27 views

Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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1answer
87 views

Why is $V^{\perp}={0}$

Let $V$ be an inner product space. I have read a statement saying $V^{\perp}=\{0\}$. Why is this true? It seems trivial to even define an orthogonal complement to $V^{\perp}$ if it is always just $0$. ...
2
votes
2answers
618 views

Proving two planes are parallel (question about the equation)

If I have two planes: $$5x + y - z = 7$$ $$-25x -5y + 5z = 9$$ I can see that from the first plane I get the vector $\langle5,1,-1\rangle$ from the coefficients and then from the second plane I get ...
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0answers
29 views

Complement of the radical of a bilinear form

The following is given(V is a vector space and $\gamma$ is a bilinear form: $rad(V)=\{v\in V|\;\gamma(v,w)=0 $ for all $ w \in W\}$ Let U be the complement of rad(V), show that $V=(U,\gamma_{|u})\...
9
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1answer
103 views

Continuously extending a set of independent vectors to a basis.

Question: Let $I=(a,b)$ be an interval and let $$v_i:I\to\mathbb{R}^n,\quad i=1,\ldots,k$$ be continuous curves such that $v_1(t),\ldots,v_k(t)$ are linearly independent in $\mathbb{R}^n$ for ...
2
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1answer
127 views

Prob. 14, Sec. 2.10 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: application to a system of equations?

Let $M$ be a non-empty subset of a normed space $X$, and let $M^a$ denote the subspace of the dual space $X'$ that consists of all those bounded linear functionals that vanish at each point of set $M$....
3
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1answer
93 views

New Horizons at Pluto

I recently posted this question on the signal processing site http://dsp.stackexchange.com/questions/23768/new-horizons-at-pluto The only answer was less detailed than I hoped for, so I'm trying here ...
2
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3answers
33 views

$\sum^6_{i=1}(x_i-\bar{x})^2$ as $\sum^6_{i=1}x_i^2 - 6\bar{x}^2$ what rules where applied?

consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$. How would I transform the first term by hand ...
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1answer
69 views

A question in matrix norm. [duplicate]

Let $A \in {M_n}$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why does ${\left\| {\left| A \right|} \right\|_2} \le \left\| {\left| A \right|} \right\|_1^{\frac{1}{2}}\left\| ...
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0answers
40 views

Solving simultaneous recurrences

I've been reading about characteristic equations for recurrence relations and I was wondering how one would solve a simultaneous recurrence, such as $$f(n) =c_1g(n-1) +c_2f(n-1)+ c_3$$ $$g(n) = d_1f(...
1
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1answer
48 views

Verifying if these basis are positive or negative?

Verify if the basis $E=(e_1,e_2,e_3)$ and $F=(f_1,f_2,f_3)$ are positive or negative with: $$f_1=e_1\quad \quad\quad\quad\quad f_2=e_2+e_3\quad \quad \quad\quad \quad f_3=e_1+e_2 $$ I did the ...
1
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1answer
53 views

Approximating matrix with $n>1$ rank as outer product of vectors

I know that matrix $M$ can be represented as outer product of two vectors (lets say $x$ and $y$) if it is of rank 1. Is there any way of approximating vectors $x$ and $y$ such that $|M - x * y|$ is ...
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0answers
18 views

Simplification of Bubnov-Galerkin

In my textbook about advanced discretization methods it is stated that the Bubnov-Galerkin method can be simplified. $\mathcal{S}^h=\left\lbrace v^h+q^h\left|v^h\in\mathcal{V}^h\right.\right\rbrace$, ...
6
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1answer
329 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
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0answers
27 views

Solving Coupled equations

I need to solve a coupled equation and basically I am completely stuck on how to proceed. The equations are $$ a = u_\pm + \frac{i}{b}\cdot\frac{u_+ - u_-}{\sqrt{u_{\pm}^2-1}} $$ and $$ N_\pm(a) = ...
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4answers
37 views

Integers $a$ for which the equation $\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$ has the most solutions

$$\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$$ I know this should be done graphically, looking at each case and seeing for which $a$ will it intersect the $x$-axis the most times, but I can't seem ...
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1answer
71 views

Positive linear combinations of intervals

Given two intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1=1-a_0<1$ and a third interval $I=[-a,a]$ where $0<a<\frac{1}2$, when is there an $\alpha,\beta\in\Bbb R$ such that ...
3
votes
5answers
715 views

If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis

If $V$ is a vector space and $v_1, v_2, . . . , v_n \in V$ span $V$, and $u_1, u_2, . . . , u_m ∈ V$ are linearly independent, then $m\le n$. Use this to prove that if $V$ has dimension $n$ and $u_1, ...
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1answer
50 views

Adjoint of a linear operator

Let $V$ be an inner product space. We define a linear transformation T as $T(\alpha)=(\alpha|\beta)\gamma$, where $\beta$ and $\gamma$ are fixed. Then show that this operator has an adjoint. Find it ...
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1answer
45 views

Consider the linear transformation $L:R^n\to R^n$ defined by $L\left(X\right)=AX$, then $A$ is diagonalizable iff the matrix of $L$ is diagonal.

I was asked to study the following corollary I could only understand up to theorem 3, does anyone know what the name of this corollary is and if there is clearer proof online?
3
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1answer
53 views

Given W with a basis, how do I find the basis for orthogonal complement of W?

Let $V$ be the inner product space of $P_{3}$ and $W$ with basis $\{1, \mathbb t^2 \}$. Find a basis for $W^\perp$. I'm not quite sure what I need to do here first. Do I need to use Gram-Schmidt? ...
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1answer
63 views

Three circles each pair of which intersects at two points

consider three circles in euclidean plane $\mathbb{R}^2$ each pair of which intersects at two points. Therefore any two circles determine one line (going through those two points where the circles ...
2
votes
4answers
116 views

The trace identity $\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB)$

Prove that $$\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB).$$ Else show a counterexample. I've tried using the trace properties such as $$\text{tr}(A+B) = \text{tr}(A) + \...
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0answers
64 views

Eigenvector Sensitivity

The Maximum Likelihood Estimator in a model I am studying depends on the eigenvectors of a symmetric positive definite (covariance) matrix ${\bf{X}}$ (3000 * 3000). I believe that the answer ought to ...
2
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1answer
405 views

Dimension of set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$

Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is (A) $\left(\begin{matrix}n\\d\end{matrix}\right)$ (B) $\left(\...
5
votes
1answer
71 views

Linear Transformation Matrix with polynomials

A linear transformation $T : P_2 \to P_2$ has matrix with respect to $S$ given by: $$[T]\,( S) = \begin{bmatrix} 1/2&-3&1/2\\ -1&4&-1\\ 1/2&2&1/2\\ \end{bmatrix} $$ How do ...
4
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1answer
67 views

Determinant defined using multilinear alternating maps, and invertibility of linear endomorphisms

In Jeffrey Lee's differential geometry text on page 353 he defines the determinant in an interesting way using multilinear alternating maps: Suppose $V$ is an $n$-dimensional $k$-vector space over ...
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0answers
25 views

Proof of direct sum with span and basis [duplicate]

Let $v_1, v_2,\dots, v_n$ be a basis of the vector space $V$. Let $k$ be an integer such that $1 \leq k \leq n$, and put $U = \operatorname{Span}\{v_1, . . . , v_k\}$ and $W = \operatorname{Span}\{v_{...
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1answer
41 views

Find all solutions for $B$ of $AB=H$ where $A,H$ are rectangular matrices of a given rank

Let the size of the a matrix $A$ be $n\times m$ and $n\times p$ of a matrix $H$ and assume $\mathrm{rank(A)}=\mathrm{rank(H)}=n$ and $m>p>n$. How do I find all possible solutions for $B$ (whose ...
4
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1answer
156 views

Geometric description of linear equation

Consider the system of equations. Give a geometric description of the intersection of the three planes when k=2 and k=0 eq.1) $x+2y-z=-3$ eq.2) $3x+5y+kz=-4$ eq.3) $9x+(k+13)y+6z=9$ ...
2
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0answers
75 views

Proving a kernel matrix is positive semidefinite

Given $m$ vectors $x_1, x_2,...,x_m$ in n dimensional boolean space. I am trying to prove that gram/kernel matrix $G$ is positive semidefinite, such that $$G_{ij} = \frac{<\vec x_i,\vec x_j>}{&...
4
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1answer
52 views

Tensor, exterior, symmetric powers over fields of nonzero characteristic

I was reading Fulton and Harris' discussion of exterior and symmetric products as quotient spaces of tensor products in their rep theory book when I noticed that they made this claim (the emphasis is ...
5
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2answers
279 views

Is there any inverse-commutator for matrices

My question is very simple. Given a symmetric real matrix $A$, and a square real matrix $C$, how can one solve the equation $[A,X]=C$, where $[A,B]$ is commutator of $A$ and $B$, i.e., $[A,B]=AB-BA$. ...
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2answers
97 views

Show that $T$ is a linear operator - Linear Transformations in Linear Algebra

We are asked: Consider the operator $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ where $T(x_1, x_2) = (x_1 + kx_2, -x_2)$ for every $(x_1, x_2) \in \mathbb{R^2}$. Here $k \in \mathbb{R}$ is fixed. a) ...
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1answer
41 views

Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
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0answers
64 views

Are Gaussians a basis for the vector space of continuous functions?

How can I prove (or disprove) that the Gaussian function family: $f_{\mu,\sigma}(x)=e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$ Are a basis for $C(\mathbb{R})$ ?
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1answer
15 views

Finding subsets of all the 3-dimensional vectors over finite field, such that every triple is independent

If you have a set of points $P = \{(x_i, y_i) : 0 < i < k\}$ in the general position on the plane (no three collinear), then the matrix constructed by having the $i$th row ($R_i$) equal to $(x_i,...
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1answer
133 views

What does the “$\cdot$” mean in an equation [duplicate]

I am trying to solve an equation for a project that I am undertaking. The equation is very long and its probably not necessary to show it all here. Most of the equation is fairly straightforward; i.e.,...
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2answers
27 views

Finding vector which have a certain norm.

Let $$A=\begin{pmatrix} 9 & 5\\ 11 & 6 \end{pmatrix}$$ Find the conditionnumber $K(A)$ (for $\|\cdot \|_\infty$), and then try to find $b, r \in \mathbb{R}^2$ such that (for $Ax= b$ and $...
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1answer
17 views

Basis of tensor product of two vector spaces

I'm doing a proof of just two spaces, so $V^* \otimes W^*$ has basis $$\{\epsilon^{(1)}_{i_1} \otimes \epsilon^{(2)}_{i_2} \mid 1\leq i_1 \leq n_1,1\leq i_2 \leq n_2\}$$ For any $w_1\otimes w_2$ in ...
2
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1answer
45 views

Finding a Dual Sequence for a Sequence of Polynomials

I am reading a paper on quadratic decomposition of Appell Sequences and would like to see if I can apply it to a particular Appell sequence that I am working with. However, my undergraduate Linear ...
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2answers
35 views

$\frac{1}{n} \left( \sum_{i=1}^n a_{ii} \right)^2 \le \|(a_{i,j})\|^2_F$

I want to show that one can estimate the Trace of a matrix by the Frobenius norm of a matrix. $$ \frac{1}{n}\left( \sum_{i=1}^n a_{ii} \right)^2 \le \|(a_{i,j})\|^2_F.$$ Unfortunately, I think that ...
1
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1answer
24 views

Problem about tensor powers of operator

$F_1,\dots,F_m$ - linearly independent endomorphisms of vector space $V$. I want to prove that there is no $\lambda_1,\dots,\lambda_m$(not all $\lambda_i = 0$) such for any $n > 1$ $$\lambda_1 F_1^{...
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1answer
91 views

Operation research - postoptimality analysis - find all solutions to problem

I'm currently learning Operations Research from "Introduction to Operations Research - Hillier". I know that somethimes a problem has many optimal solutions. For example in a two dimensional problem ...
3
votes
1answer
36 views

Finding a basis for the nullspace

$A$=\begin{bmatrix}-2 & 5 & 3 & -1\\ 0 & 1 & -4 & 2\\ 6 & -14 & -13 & 1\\ 0 & 0 &0 &0\end{bmatrix} I need to find the null space for this matrix. After ...