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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17 views

Quention about the historical definition of determinant

$$ax+by = k_1\\cx + dy = k_2$$ If I want to solve for $y$ in the first equation: $$by = k_1 - ax\implies y = \frac{k_1-ax}{b}$$ Then substitute $y$ in the second equation: $$cx + d\frac{k_1-ax}{b} ...
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1answer
17 views

Name for problems where the constraints are on inner products

I have a problem with a lot of dot-product constraints like $V_1 \cdot V_2 = 0$ or $V_1 \cdot V_3 = V_2 \cdot V_4$. However, I don't know what these types of problems are called so I can't look up ...
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1answer
30 views

Projective Geometry in $\mathbb{R}^{3}$: “Lonely lines” in source/image planes

I am reading some lecture slides about projective geometry in $\mathbb{R}^{3}$. In particular, given a source plane, $S$, an image plane, $I$, and a focal point, $f$, the issue at hand is the ...
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1answer
21 views

Trouble showing spans of two bases are equivalent

I was given the following problem: Let V be a vector space over field F. Show that x,y $\in$ V form a basis iff x+y, x-y form a basis. But I seem to be stuck when showing the span of one basis ...
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1answer
26 views

Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
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2answers
25 views

Determinant by nullifying

I am supposed to calculate the value for the determinant of this matrix. I didn't know what to do, so I looked up for the sample solution, which I don't understand. $$\left|\begin{array}{ccc} 18 ...
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0answers
17 views

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, Prove that $|(A+A^T)(B+B^T)|=0$

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, prove that $\det[(A+A^T)(B+B^T)]=0\ \ $ with $ \ k\in \mathbb{N}$ I don't have ideas for this problem. Thanks !
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9 views

Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
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1answer
12 views

Prove or disprove isomorphism problem

P is a 2*4 matrix, which has rank (P) = 2, L: M 4*4 -> M 2*2 is a linear mapping, defined by L(A) = P A P^T, ---(PAP transpose). I can see that L is not one-to-one, as A must be in the null-space of ...
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12 views

Representing commuting operators as functions of a third operator.

Let A and B be commuting operators, then there is a maximal operator C and there are functions f, g such that A = f(C) and B = g(C). I'm looking for a proof of this theorem. I don't fully understand ...
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2answers
21 views

Finding a Basis for polynomial subspace

This is problem 14 in Herstein's Topics in Algebra. I'm having trouble with the problem (working through the text independently). For $F$ a field, define $V_n=\{p(x)\in F(x) : \deg p(x)<n, n\in ...
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0answers
18 views

Ker and Im sum of matrix [on hold]

Suppose we have matrix and we have found Im and Ker as vectors.How to find Im+Ker?
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1answer
65 views
+100

Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
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2answers
19 views

How is the Set of all Polynomials Equal to the Following Union?

Given that $P(F)$ is the set containing all polynomials with coefficients from field $F$, I am given the following: $W_1$ is the set of all polynomials $f(x)$ in $P(F)$ such that for: ...
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0answers
17 views

Geometry of Spans in $\Bbb{R}^2$ and $\Bbb{R}^3$

I'm having difficulty figuring out how to approach the following Geometry of Spans questions. I only seem to understand the "span of a single vector" ones. How would I go about explaining the others? ...
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0answers
9 views

Proving that the derivative of the LRL vector $=0$ [on hold]

How to prove that the derivative of the Laplace–Runge–Lenz vector $=0$? $$A=\dot{x}\times(x\times\dot{x})-\dfrac{k}{\mu}\cdot\dfrac{x}{||x||}$$ $$\dot{A}=0$$
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3answers
41 views

Relationship between Eigenvalues

I am looking at a matrix $$\mathbf{M} = \left(\mathbf{I}+k\theta\mathbf{B}^{-1}\mathbf{A}\right)^{-1}\left(\mathbf{I}-k(1-\theta)\mathbf{B}^{-1}\mathbf{A}\right) $$ where $\mathbf{I}$ is the identity ...
1
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1answer
18 views

Sequence forming a vector space

The sequences $(x_k)_{k=1}^{\infty}$ in $\mathbb R$ , all or almost all $\neq 0$ with operations defined component by component, form a vector space V over $\mathbb R$. Find a basis of V, ...
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1answer
26 views

Linear algebra homogenous system

Given a $3\times3$ matrix depending on a real parameter $x$. Denote by $S(A(x))$ the space of all solutions of the homogenous system $A(x)Y=0$. How can one find this space in generally ?
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1answer
44 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
3
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1answer
35 views

Axler LADR Exercise

The exercise is: Suppose $v_1, \ldots , v_m$ is linearly independent in $V$ and $w \in V$. Prove that if $v_1+w, \ldots, v_m+w$ is linearly dependent, then $w \in \operatorname{span}(v_1, \ldots, ...
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1answer
22 views

Direct-sum of subspaces

Let $V$ be a finite-dimentional vector space and let $W_{1},\ldots, W_{k}$ be subspaces of such that $V=W_{1}+\ldots+W_{k}$ and $dimV=dim W_{1}+\ldots+dim W_{k}$ Prove that $V=W_{1}\oplus\ldots\oplus ...
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1answer
33 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
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2answers
26 views

How do you solve this kind of homogeneous linear system?

Suppose the matrix associated with a homogeneous linear system is \begin{pmatrix} -31&0&0&4\\-8&0&1&-1\\0&0&0&0\\-4&0&-2&-1\end{pmatrix} How do you ...
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0answers
16 views

Show that the subspace $B_1$ is a basis of $C^4$

I have $B_1$ = $((i,0,0,0),(1,0,1,i),(0,2,i,0),(-i,0,0,i))$ And $C^4$ is a vector space and a basis of it is $C_b$ = $(e_1,e_2,e_3,e_4)$ I want to show that $B_1$ is a basis of $C_b$. So i introduce ...
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0answers
22 views

Show Projection minimizes variance

Van der Vaart's Asymptotic Statistics, problem 11.2 Another idea of projection is based on minimizing variance instead of second moment. Show that $\text{Var}[T-S]$ is minimized over a linear space ...
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0answers
77 views

Gradient function for restricted likelihood with respect to terms that influence Sigma

Is there a straightforward/generalized way to calculate partial derivatives of the restricted multivariate log-likelihood function $\ln\mathscr{L}=C+\ln\lvert ...
2
votes
1answer
21 views

Inner Nilpotent Derivation

Some context first: Consider $S=M_n(\mathbb{C})$ as an algebra over $\mathbb{C}$. For every $A \in S$, it's easy to check that $ad_A(M):=AM-MA$ is a derivation ($C$-homomorphism of $S$ that satisfies ...
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4answers
162 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
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24 views

A question about the vector space of Fibonacci sequences

Question: Let $V$ be the vector space of real sequences over $\mathbb{R}$. If $W$ is the subspace of all Fibonacci sequences (i.e. a sequence $\{a_n\}\in W$ if $a_n=a_{n-1}+a_{n-2}$, for all $n\geq ...
0
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1answer
11 views

Respresting linear transformation with matrix with restrictions

When given a set of restrictions, what is the way to find a representing matrix of a linear transformation? Lets say I have T:R^4->R^3 and I need the Ker(T) to be spaned by {(1,2,3,4), (0,1,1,1)}. ...
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2answers
25 views

U=span{(1,1,-1), (2,3,-1), (3,1,-5)} and V=span{(1,1,-3), (3,-2,-8), (2,1,-3)}. What is U $\cap$ V

Let U=span{(1,1,-1), (2,3,-1), (3,1,-5)} and V=span{(1,1,-3), (3,-2,-8), (2,1,-3)}. What is U $\cap$ V? 1. U 2. V 3.zero subspace 4. None of these. I tried firstly to find dim of U $ \cap$ V , by ...
5
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2answers
135 views

For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
2
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0answers
26 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
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2answers
21 views

Show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$ for a normal matrix $A$ with an eigenvalue $c$

Suppose $A \in M_{n\times n}(\mathbb C)$ is a normal matrix and $c$ is an eigenvalue of $A$. I'm trying to show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$. I know that if we were ...
3
votes
2answers
109 views

How to show that $A^2=AB+BA$ implies $\det(AB-BA)=0$ for $3\times3$ matrices?

Let $A$ and $B$ be two $3\times 3$ matrices with complex entries,such that $A^2=AB+BA$. Prove that $\det(AB-BA)=0$. (Is the above result true for matrices with real entries?)
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4answers
243 views
+100

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
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1answer
23 views

Misunderstanding in the proof that the sum of subspaces is the smallest containing subspace.

So if $V_1,...,V_n$ are subspaces of $M$ then $V_1+...+V_n$ is the smallest subspace of $M$ containing $V_1,...,V_n$ The proof is that clearly $V_1,...,V_n$ are all contained in $V_1+...+V_n$ Then ...
3
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2answers
989 views

Symmetric Tridiagonal Matrix has distinct eigenvalues.

Show that the rank of $ n\times n$ symmetric tridiagonal matrix is at least $n-1$, and prove that it has $n$ distinct eigenvalues.
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1answer
21 views

Linear least-squares with matrices rather than vectors

I have two coordinate frames, each represented by a 4-by-4 matrix ($A$ and $B$), where this is the pose (orientation and translation) in homogeneous coordinates. I now want to find a third matrix $T$, ...
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1answer
58 views

An effcient method of solving a Diophantine equation with 3 variables $Ax+By+Cz=D$?

I'm trying to make an efficient algorithm to find one of the solutions and how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb ...
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1answer
70 views

The set of differentiable functions such that $f'(2)=b$ is a linear subspace if and only if $b=0$??

Questions are in bold. The set of differentiable real-valued functions on (0,3) such that $f'(2)=b$ is a subspace of $(0,3)\to \mathbb R$ if and only if $b=0$ ($(0,3)\to \mathbb R$ denotes the set of ...
-1
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1answer
11 views

Finding base B'

I have B = {(0,2,1),(-2,2,1),(-1,2,1)} how can I find B' so $ x + [x]_B + [x]_{B'} = 0 $ (equlas zero vector). For every vector $ x \in \mathbb{R}^{3} $.
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2answers
349 views

Finding the inner product generated by a matrix

In each part, use the given inner product on $R^2$ to find $\|\vec w\|$, where $\vec w\ = (-1, 3)$. Then the problem lists different inner products to use to find the norm but the one I'm ...
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2answers
31 views

Multiplication of rational with irrational number?

If $a$ is rational and $b$ is irrational number and we know that $ab$ is rational, then what can we say about $a/b$? Is true that it's equal to 0?
3
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3answers
50 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
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1answer
25 views

Define: A solution of a linear equations system + Row, Column & Null spaces relations

The linear equations system: $$\left(\begin{array}{ccc|c}1 & 1 & 1 & 3 \\1 & 2 & 3 & 6 \\1 & 3 & 5 & 9\end{array}\right).$$ Has the following solution: $$ ...
1
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1answer
24 views

Linear Transformations on Infinite Dimensional Vector Spaces

Let $T$ be a linear transformation $T:V\to V$, where $V$ is an infinite dimensional vector space. How can we construct examples such as $1.$ T is one to one but not onto $2.$ T is onto but not ...
3
votes
1answer
42 views

Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a complex unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), that is, $-1$ is not an eigenvalue of $Q$, then $$ A:=(I-Q)(I+Q)^{-1} $$ is ...
0
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0answers
9 views

Diagonalization of a quadratic form with parameter $k \in \mathbb{R}$: $q(x,y,z)=(2+k)x^2+2y^2+kz^2+4xy-2kxz$

Let $q: \mathbb{R^3} \to \mathbb{R}$ be the quadratic form $$q(x,y,z)=(2+k)x^2+2y^2+kz^2+4xy-2kxz,$$ with $k \in \mathbb{R}$. I would like to diagonalize this form and then write it in the canonical ...