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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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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8 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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1answer
10 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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2answers
18 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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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
16 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
21 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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14 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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2answers
17 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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8 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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1answer
17 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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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 ...
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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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1answer
34 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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2answers
18 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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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
20 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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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 ...
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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
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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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 ...
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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 ...
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1answer
31 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
108 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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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
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?
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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: $$ ...
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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 ...
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2answers
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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 ...
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1answer
20 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
23 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 ...
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1answer
48 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
3
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1answer
41 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 ...
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1answer
18 views

Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
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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 ...
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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Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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2answers
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Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
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2answers
99 views

Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
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1answer
18 views

Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional

I am having trouble understanding this whole question, and how to prove it. Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of ...
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how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
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1answer
51 views

set of all $2\times 2$ matrcies having neither eigen value is real

Could any one tell me whether the following subsets of $M_2(\mathbb{R})$ are open, closed or neither open nor closed? set of all $2\times 2$ matrcies having neither eigen value is real. set of all ...
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3answers
23 views

$Ker(T) \subseteq Ker(S)$ implies the exist some $H$ s.t $H\circ T=S$

Let $V,W$ be a vector space over $\mathbb{F}$, with finite dimension. Let $T,S:V\rightarrow W$ linear transformations such that $Ker(T)\subseteq Ker(S)$. Prove that exists some linear transformation ...
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2answers
56 views

Solving a system of three simultaneous equations

Given the system $$ \begin{align*} -2x + ay - bz &= -4 \\ x + bz &= 2 \\ 2x + y + 3bz &= b \end{align*} $$ The question asks to find conditions on $a$ and $b$ that the system has no ...
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1answer
29 views

Property of orthogonal and skew symmetric matrix

If $A$ be a $n\times n$ orthogonal matrix and $X$ be a matrix such that $X=(A+I)^{-1}(A-I)$ then show that $X$ is a skew-symmetric matrix,whenever $n$ is an odd integer.
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1answer
19 views

Kernel and image of a diagonalizable endomorphism $f$ given only an orthogonal basis$B={w_1,w_2,w_3}$, an eigenvalue, and that $f(w_1)=f(w_2)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be diagonalizable with $B$, basis of eigenvectors, such that $B={w_1,w_2,w_3}$, where $w_1=(1,2,0),w_2=(0,1,1),w_3=(0,1,-1)$. If we know that $3$ is an ...
2
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0answers
27 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix in $\mathbb{C}$ $$U = \{a_{ij}\}_{n\times n},$$ where "unitary" means $$U'U = I, \quad U'\text{means conjugate transpose.} $$ I know how to calculate its degree of ...
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2answers
27 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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3answers
50 views

I have difficulty understanding functions forming vector space.

I have knowledge of basic linear algebra, so I can understand the finite vector space as linear combinations of vectors of $R^n$. However, when it comes function as vector and functions form a ...