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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6
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The determinant of adjugate matrix

I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
3
votes
2answers
493 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
0
votes
1answer
23 views

A question about Linear transformations on matrices

We know that if $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear transformation, then there exists a matrix $A\in \mathbb{R}^{m\times n}$ such that $T(x)=Ax$, for every $x\in\mathbb{R}^n$. Now, if ...
2
votes
1answer
2k views

Easy way to calculate inverse of an LU decomposition.

I have a matrix A and a lower triangular matrix L (with 1's along the diagonal) and an upper triangular matrix U. These are constructed such that $A=LU$. I know that $A^{-1} = L^{-1}U^{-1}$ and I know ...
0
votes
0answers
10 views

Gell-Mann Matrices with dot and cross product

We know that, given $\vec{a}$ and $\vec{b}$, then $(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma})=(\vec{a}\cdot\vec{b})\mathbb{I}+i(\vec{a}\times\vec{b})\cdot\vec{\sigma}$. $\sigma_i$ are the ...
0
votes
3answers
53 views

Is there any relatively quick way to diagonalize this matrix with an orthogonal matrix?

3a of this released exam asks (paraphrased): Diagonalize the matrix $$A = \begin{bmatrix} 0 & 4 & 0 \\ 4 & 0 & 4 \\ 0 & 4 & 0 \end{bmatrix}$$ by an ...
15
votes
5answers
1k views

Simple linear algebra problem: prove a matrix is invertible

I'm preparing for a test in linear algebra and I've come across a problem I'm having trouble with for some reason: Given a square matrix A, $A^2=2I$, prove that $A-I$ is invertible. I know this is ...
0
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0answers
18 views

Does anyone know a reference to best-fitting lines with integral coefficients?

I'm writing up a manual on how to generate "nice" Linear Algebra problems; that is, where the solutions tend to be integral. I "discovered" the following fact about the best-fitting line: Theorem. ...
0
votes
1answer
21 views

Integrate this Gaussian in $\mathbb{R}^N$

I'm trying to compute this integral: $$\int_{\mathbb{R}^N} \exp\Big((\vec{x} - \vec{\mu})^T(\lambda \text{A}^T \text{A}+\delta \text{L})(\vec{x} - \vec{\mu}) + U(\delta)\Big)d\vec{x}$$ Where $$\mu = ...
0
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0answers
18 views

Halmos or Axler for a healthy course in linear algebra.

I am reading linear algebra from the book by I.N.Herstein and following the book "Finite dimensional vector spaces" by Halmos as reference. I am asking if these books are quite complete or I should ...
3
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0answers
41 views

ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

That is we have a linear transformation, i.e. an $ n\times n $ matrix $A$, such that for every pair of vectors $ v $ and $ w $ we have $$ \langle v,w\rangle=0 \ \ \ \implies \ \ \ \ \ \langle ...
1
vote
1answer
16 views

Differentiation map

When we study linear homomorphism. We have an example of surjective of a linear homomorphism as follows: $V:=K[x]$, $K$ is an arbitary field and the linear map $\bf{T}$ as follows $\begin{aligned} ...
0
votes
1answer
82 views

Proving that $UT=TU$ iff $U=g(T)$ for some polynomial $g$

Let $T$ be a linear operator on a vector space $V$, which is a $T$-cyclic subspace of itself. How do I prove that $UT=TU$ iff $U=g(T)$ for some polynomial $g$?
1
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1answer
38 views

Does there exist a subspace $W\subseteq M_{4\times 4} (\mathbb{R})$, such that the length of the orthogonal projection of a certain matrix is 5?

Let $A= \left( \begin{array}{ccc} 1 & -1 & 1 & -1 \\ -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 \\ -1 & 1 & -1 & 1 \\ \end{array} \right) $ a matrix in ...
1
vote
1answer
223 views

Let V be an n-dimensional vector space, and T from V to V a linear transformation…

Let $V$ be an $n$-dimensional vector space, and $T$ from $V$ to $V$ a linear transformation. Show that $T$ is nilpotent of order $n$ if and only if there exists a basis $\beta = {v_1,v_2,....v_n}$ of ...
1
vote
2answers
54 views

Find the standard matrix representation of The composite linear transformation [closed]

Find the standard matrix representation of the composite linear transformation: yaw of 90 degrees, pitch of 45 degrees, roll of -45 degrees.
2
votes
2answers
230 views

3D rotation around arbitrary axis

I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and ...
0
votes
1answer
21 views

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
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votes
3answers
30 views

Null Space of an Identity Transformation

Let $V$ and $W$ be vector spaces and let $I: V \to V$ be the identity transformation. In Linear Algebra by Friedberg, Insel and Spence, it states that the nullspace of an identity transformation is ...
1
vote
1answer
15 views

Why is the representation of a vector with respect to basis an isomorphism?

As part of a proof on whether any matrix represents a homomorphism between vector spaces of the appropriate dimensions with respect to any pair of bases, my professor mentioned that for the map h:V to ...
0
votes
0answers
19 views

Modeling simple linear equations

This should be pretty simple but I'm blanking on this. I need to model (graph) how path 1 becomes equally as efficient as path 2 as the distance of path 2 increases. distance of path 1 (from A to B) ...
0
votes
2answers
23 views

Given a Matrix A, prove that 1/9A is an orthogonal matrix.

$$Let A= \begin{pmatrix} 4 & -7 & 4 \\ -1 & 4 & 8 \\ -8 & -4 & 1 \\ \end{pmatrix} $$ The problem is to prove that $1/9A$ is an ...
1
vote
1answer
31 views

Is a one-dimensional vector space orthogonal?

I'm working on an assignment where we need to orthonormalize bases, but one of the problem is a one-dimensional subspace of R3. I know the definition for orthogonal relies on the scalar product, but ...
0
votes
5answers
61 views

Solving equations.

How would you solve these equations and show that they do not intersect each other? $$x^2+y^2=2x-2y$$ $$x^2+y^2=4(x^2+y^2)^{1/2} +y$$ It's isolating a term which I am struggling with. General ...
1
vote
1answer
389 views

Kalman filter and data extrapolation

Context of the situation: I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
0
votes
1answer
395 views

How do I find transformation matrix with respect to given basis in the domain and/or the codomain, given the transformation in the standard basis?

I´m being given a linear transformation, for which I can find the standard basis in the domain and codomain; but then, the book ask to find the associated matrix related to a new basis for the ...
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votes
0answers
21 views

question on linear algebra word problems. [on hold]

How would you go about answering this problem I tried this: Wayside Auto Sales discovers that when \$1000 is spent on radio advertising, weekly sales increase by \$101,000. When \$1250 is spent on ...
0
votes
1answer
11 views

Formula for calculating markup with big % for small amounts and small % for larger amounts

I am trying to come up with a formula for calculating markup for products that range in value from a few cents up to tens of Dollars. At 10c I would like the markup to be around 500%, and from 2 ...
-1
votes
1answer
13 views

Linear functions and equations in text

The question is: A rectangle has a width of 28cm. The one side is 2cm longer than the other side. We are supposed to form an equation and then solve it, but I don't know to form an equation ...
0
votes
1answer
27 views

Let V be an inner product space.Then for x,y,z belongs to V and belongs to field,F,the following statements are true.

(a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$. (b) $\langle x,cy\rangle =\bar c\langle x,y\rangle$. (c) $\langle x,0\rangle = \langle 0,x\rangle =0$. (d) $\langle x,x\rangle=0$ ...
-2
votes
0answers
20 views

Would these equations be correct or are they false? [on hold]

the variables for the equations are below, the point of this is getting a good percentage of $ return with Bitcoin... a = amount of $ c = amount of $ for computer r = amount of $/second N = new ...
0
votes
1answer
22 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=Id-ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix and $I_n$ the identity matrix. Let be $A = I_n-ab^T$. What is the determinant of $A$?
1
vote
1answer
21 views

The sum of $V=U+W$ of a vectorspace V and subspaces $U$, and $V$

I know what the sum of two subspaces is and how we notate but is it ok to write a minus to denote what I hope should be obvious is meant. So we have $V=U+W+Y$ where $V$ is a v.space and $U,W,Y$ ...
1
vote
2answers
31 views

why nullspace is the largest subspace perpendicular to the row space?

The proof from my textbook is "If x were a vector orthogonal to the row space, but not in the nullspace, then the dimension of $C(A^T)^\perp$ would be at least n — r + 1. But this would be too large ...
1
vote
2answers
32 views

Problem in solving a question of vector space.

The question is : Let, $V$ be the subspace of all real $n \times n$ matrices such that the entries in every row add up to zero and the entries in every column also add up to zero. What is the ...
1
vote
2answers
52 views

Does every invertible matrix A has a matrix B such that A=Adj(B)?

I'm trying to understand if it's always true, always true over $\mathbb C$ or never true. I know that if $A$ is invertible, than there exists $A^{-1}$. $$A=\frac{1}{det (A^{-1})}Adj(A^{-1})$$ So I ...
1
vote
1answer
12 views

There is only one linear function whose image of a specific base is a specific set of vectors.

So I found this theorem: Let $V$ be a real vector space of dimension $n$, and $\mathscr{B} = \{\mathbf{b}_1, \dotsc, \mathbf{b}n\}$ its base; let $V'$ be a real vector space and $\mathbf{c}_1, ...
4
votes
2answers
117 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
39
votes
12answers
68k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
0
votes
1answer
44 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
1
vote
1answer
32 views

Decomposing vector space into positive/negative definite subspaces

Consider the quadratic form: $$f:\mathbb{R}^3\to\mathbb{R};\quad (x,y,z)\mapsto x^2+2y^2-2xy-2xz$$ I am doing a problem which asks me to find subspaces $A,B\subseteq \mathbb{R}^3$ such that ...
0
votes
3answers
45 views

find a vector perpendicular to set of vectors

I have a set of $m$ vectors given by $$A =\left(\vec{v}_0,\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\right),$$ elements of which need not be all linearly independent, in an $n$-dimensional space. I want a ...
7
votes
1answer
81 views

The lattice points in the real cone of some semigroups are just the integer cone of that semigroup.

I'm trying to solve an exercise in Fulton's book on toric varieties, and have reduced it to the following: Let $M$ be a lattice of rank $n$ with $M \otimes \mathbb{R} = V$, and $S$ be a finitely ...
0
votes
2answers
55 views

Intersection of 2D planes in 4D space

If I had a four dimensional space, in which I embedded two planes, what possible intersections could they have? Constructing a Plane To give this more context, consider the following. If I had a 4d ...
3
votes
3answers
37 views

Show that $\dim(\operatorname{range}(T)) = 1$.

Let $T :\mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation such that $T \neq 0$ but $T^2=0$. Show that $$\dim(\operatorname{range}(T))=1$$
0
votes
0answers
38 views

identity for quaternions' group Sp(n)=Sp(2n,C)∩U(2n)

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
7
votes
2answers
2k views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
0
votes
1answer
13 views

SVD and homogeneous equation

Suppose a $m \times n$ matrix $A$, and column vector $h$. ($A$'s rank is equal or smaller then $n$(=$h$'s length).) If, $$ Ah=0 $$ then $h$ can be the last column of $V$ where $A = UDV^T $. ...
1
vote
2answers
24 views

Determining similar matrices

I have this matrix $$A= \begin{bmatrix}1 &0& 2\\0&-1&-2\\2&-2&0\end{bmatrix}$$ I found the eigenvalues to be $0, 3, -3$ I am tasked with finding if $A$ is similar to a ...
0
votes
0answers
28 views

Derivative of $\frac{dA^TA}{dx}$ [duplicate]

I think I have an easy question but I can't understand how to do. I am trying to figure out how to do the derivative of a symmetric matrix $B = A^TA$ w.r.t. one parameter $x$ of the matrix: ...