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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11 views

Where can I find linear algebra described in a pointfree manner?

Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the ...
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1answer
17 views

Change of Basis for 2x2 matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
0
votes
1answer
19 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
-1
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2answers
18 views

how many jelly beans did each girl have at first?

Martha and Mary had $375$ jelly beans in all. After Mary ate $24$ jelly beans and Martha ate $\frac 17$ of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did ...
0
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0answers
20 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} ...
0
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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 ...
-1
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1answer
29 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 ...
0
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1answer
22 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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0answers
12 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 ...
0
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0answers
18 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
13 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 ...
0
votes
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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0answers
18 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 ...
0
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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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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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2answers
20 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: ...
0
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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 $=0$? $$A=\dot{x}\times(x\times\dot{x})-\dfrac{k}{\mu}\cdot\dfrac{x}{||x||}$$ $$\dot{A}=0$$
1
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1answer
19 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, ...
0
votes
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 ...
0
votes
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 ...
3
votes
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, ...
0
votes
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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votes
1answer
23 views

Direct-sum of subspaces [on hold]

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 ...
2
votes
1answer
22 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 ...
0
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0answers
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
votes
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 ?
0
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1answer
12 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)}. ...
2
votes
0answers
27 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 $L = -DN \circ Df^{-1}.$ ...
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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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} $.
1
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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?
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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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 ...
0
votes
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 ...
1
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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
26 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 ...
2
votes
1answer
49 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
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
votes
1answer
19 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
votes
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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0answers
9 views

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 ...
0
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2answers
43 views

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 ...
1
vote
2answers
100 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 ...
1
vote
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 ...
1
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0answers
24 views

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}): ...