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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QR decomposition proof

Let $A\in\mathbb{M}_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$ and take the decomposition $A=QR$ with $Q\in\mathbb{M}_{m\times n}(\mathbb{R})$ a orthogonal matrix and ...
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The number of $3\times 3$ nilpotent matrices over $\mathbb{F}_q$ using the Orbit-Stabilizer theorem

The Fine-Herstein theorem says that the number of of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ is $q^{n^2-n}$. I am trying to verify this for the cases $n=3$ using the orbit-stabilizer ...
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Discrete Fourier Time Question

Assume that $x[0]=1, x[1]=1, x[2]=1, x[3]=1, x[n]=0$ for $n \geq 4$, find the DFT of $$\{x[n]\}=( x[0], x[1], x[2], x[3] )$$. My method of doing this is to use the DFT formula as defined here: ...
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1answer
347 views

Calculate equal distance between lines and points

How do I do something like this?: Consider the lines of k: x = 4 and l: y = 4x + 2, and the point A (0, 6). What is the equation of the parabola 'p' with focus 'A' and directive k? And calculate ...
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System of linear equations: get approximate solution with non-negative coefficients

I'm looking for a process or algorithm to help me with the following problem. I have the following vectors in $\mathbb R^{3}$: $$ \vec m_3 = \begin{bmatrix} 51.8\\ 2.9\\ 22.3 \end{bmatrix}, \vec a = ...
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1answer
68 views

Question on linear algebra - Determinant multiplication.

Does anybody have a "non brute" force way to prove the following for non-singular matrices A, B: det(AB) = det(A) det(B)
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Determine which values of $\lambda \in \mathbb{R}$ cause the following vectors to be a basis

I am working on the following problem. Suppose that $\{v_1, v_2\}$ is a basis of a real vector space $V$. For which values of $\lambda \in \mathbb{R}$ is $\{w_+, w_\lambda\}$ a basis of $V$, where ...
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Eigenvalues of composition of functions

I am trying to do the following exercise: Let $V$ be a $K$-finite dimensional vector space and let $f,g \in Hom(V,V)$. Define $Spec(f)=\{\alpha \in K / \alpha \space \text{is an eigenvalue of f}\}$. ...
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direct sum of vectors

$$U = \{(x,y,z,t) \in \mathbb{R}^4 | x + 5y + 4z + t = 0 , y + 2z + t = 0 \} $$ $$W = \{(x,y,z,t) \in \mathbb{R}^4 | x + z + 3t = 0, 2x-3y-4z+3t = 0\} $$ $U \oplus W = \mathbb{R}^4$? This is my ...
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16 views

Composing Linear Transformations

Hello and thank you in advance; The problem: "Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T ...
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Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that every eigenvalue of $B^2$ is the square of some eigenvalue of $B$.

I would like to ask you for some help in the following problem: Suppose that a matrix $B$ is diagonalisable over $\mathbb{C}$. Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that ...
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An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
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Isomorphism between vector spaces of linear transformations

Let $V,W$ vector spaces over the field $F$,and let $U: V\rightarrow W$ an isomorphism between them. Prove that the linear transformation $\mathcal{U}:\mathcal{L}(V,V)\rightarrow ...
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What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
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22 views

Tensor independence

Let $(e_{i})$ be a basis in $V$, $( \epsilon_{i} )$ - basis in $V^{*}$ so that $\epsilon_{i} (e_{j})= \delta_{i}^{j}$ (Kronecker delta, $\epsilon_{i} (e_{j}) = 1 \Leftrightarrow i=j$, otherwise it's ...
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Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
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1answer
17 views

Let $T:U\rightarrow V$ be a linear map and suppose that $rank(T)=dim(U)=dim(V)=n$. Show that the are bases where the matrix is $I_n$

I found this problem that I cannot solve, but I believe is quite interesting. We have to state whether the statement is true or false. Let $T:U\rightarrow V$ be a linear map and suppose that ...
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Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?

And what's the intuition here? This is question 6(c) in pset 1.2, Strang's Linear Algebgra, 4th Ed.
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Linear operator of infinite dimension

Let $T: V\rightarrow V$ a linear operator with finite dimension. If exists a linear operator $U: V\rightarrow V$ such that $TU=I$, prove that $T$ is invertible. Prove that if the ...
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Prove statement about projection (linear map)

I am working on the following problem and do am not sure how best to approach it. Let $U$ be a vector space over a field $F$ and $p, q: U \to U$ be linear maps. Assume $p + q = \text{id}_U$ and $pq = ...
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Generalized “softmax”

I'm looking for a function $f$ from $\mathbb{R}^n$ to $[0,1]^n$, similar to softmax in the sense that is satisfies these properties: $\sum_i f(x)_i = c$, where $c$ is a chosen constant (i.e., c=1 ...
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Comprehensive Linear Algebra Text

Occasionally I come across a fact from linear algebra that I have not seen before. These facts are often obscured in search engines by either introductory texts or unrelated papers, and it is ...
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A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
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3answers
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Understanding the difference between Span and Basis

I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Span ...
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1answer
19 views

$A$ is positive definite if and only if $Q$ is invertible for every choice of $Q$

Note that if $A \in M_{n \times n}$, $A^{\prime}$ denotes the transpose of $A$. I proved the following theorem already: $A$ is nonnegative definite if and only if there exists a square matrix ...
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1answer
38 views

Solve linear algebra system [on hold]

Solve the linear equations $a·x = c$ and $a×x+b = 0$ for $x$ (which you may take to have components $x_1, x_2$ and $x_3$) if a $6= 0$ and $b$ are constant vectors and $c$ is a constant scalar. How ...
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Complementary subspaces ($K$ and $L$) problem, where $K=ker(p)$ and $L=ker(q)$ with $p,q: U \rightarrow U$ linear maps.

I am struggling with solving the following question: Let $U$ be a vector space over field $F$ and $p,q: U \rightarrow U$ linear maps. Assume $p+q=id_U$ and $pq=0$. Let $K=ker(p)$ and $L=ker(q)$. ...
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I would like to ask you for a help at asking and presenting the math problems? [on hold]

how should I present it and what not to write down that you can help me ? Thank you all
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1answer
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What means to complete a pair of vectors $(w, s)$ to an arbitrary basis of $R^d$?

I found in an article this : Let $B = (b_1, b_2, . . . , b_d)$ be an orthonormal basis of $R^d$ such that $<b1, b2 >=< w,x >$ (where $< ... >$ denotes linear span). In order to ...
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How to find a set of integer vectors (of length L) such that all its subsets with size L are linearly independent?

Given a number $M\geq L$, how to find a set of $M$ vectors in $\mathbb{Z}_{\geq0}^{L}$, say $S=\{\mathbf{a}_1,\cdots,\mathbf{a}_L\}$, such that: 1-All subsets of $S$ with size $L$ are linearly ...
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1answer
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Prove result about basis of a linear map with specific properties

I am working on the following problem. Let $V$ be an $n$-dimensional vector space over $K$ and $T: V\to V$ a linear map. For $k = 1, \ldots, n$ let $x_k \in V \smallsetminus \{0\}$ and $\lambda_k \in ...
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2answers
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Prove or disprove $g \circ f $ is one-one $\to$ both $f$ and $g$ are one-one

Prove or disprove $g \circ f $ is one-one $\implies$ both $f$ and $g$ are one-one (if $g \circ f $ exists). I've got $g \circ f $ is one-one $\implies$ $f$ is one-one (If we assume $f$ is ...
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1answer
255 views

How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. ...
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3answers
232 views

Jordan Canonical form of a matrix over rationals whose all entries are 1.

How to compute the Jordan canonical form for the $n \times n$ matrix over $\mathbb{Q}$ whose entries equals to $1$.
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1answer
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Difference between orthogonal projection and least squares solution

When you find the least squares solution you solve $$A^TA = A\vec b$$ but to find the orthogonal projection into the "subspace" A, you multiply this result (the least squares solution) with the ...
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Find the number of distinct real values of $c$ such that $A^2x=cAx$

Let $$A= \begin{pmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ and $c$ be a real no. such that $A^2x=cAx$ for some non-zero vector $x$. Then the number of ...
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2answers
51 views

Help solve ${{z}^{3}}=\overline{z}$ ($z\in \mathbb{C}$) [duplicate]

Me and my friend try to solve $${{z}^{3}}=\overline{z}$$ where $z \in \mathbb{C}$. My way to solve it was: $\operatorname{cin}(\theta )=\cos(\theta)+\sin(\theta)i$ \begin{align} & z=r ...
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1answer
15 views

Finding a base from matrice subspace

$U,W$ are sub-spaces of $M^\mathbb{R}_{2x2}$ $$U=Sp\left\{\begin{pmatrix} 1 & 2\\ 4 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & -1\\ 3 & 2 \\ \end{pmatrix}, \begin{pmatrix} 1 & ...
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1answer
11 views

Kernel Principal Component Analysis (PCA)

I learn kernel PCA from wikipedia. In this article, the eigen equation is \begin{equation} N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha} \end{equation} where $\lambda$ is the eigen value, ...
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3answers
92 views

To Find the Nullity of a Linear Transformation …

If $V(\Bbb R) $ be the vector space of $2\times2$ matrices and $$M=\begin{pmatrix} 1 & 2\\ 0 & 3 \\ \end{pmatrix}$$ If $T:V(\Bbb R)\to V(\Bbb R)$ be a linear transformation defined by ...
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1answer
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Doubt on Kantorovich inequality. Equivalence of inequalities.

To prove de Kantorovich inequality (for that we suppose the matrix A symmetric and definite positive) I need to demonstrate the next exercise: Proof that $$(x^TAx)(x^TA^{-1}x) \leq \frac{(\lambda_1 ...
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2answers
198 views

Show that 1 and -1 are the only eigenvectors of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
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1answer
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prove or disprove Composition of linear transformations is one-one

Let $T$ and $F$ be 2 linear transformations from $\Bbb R^n \to \Bbb R^n $.Then prove or disprove $T \circ F=0 \to T$ is one-one. $|TF|$$=0$ $\implies$ $|T|$$|F|$$=0$ $\implies$ either $|T|$=$0$ ...
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1answer
34 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
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54 views

Find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not. Have you got any ideas of easy examples? Thank you!
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The relation between the algebraic dimensions of a vector space and its dual

Let $V$ be an (infinite dimensional) vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what ...
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Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...
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1answer
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Calculating the determinant of an interationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$, $R$ is upper triangular, $L$ is lower ...
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1answer
40 views

Determine whether the following set is a vector space

Being pretty new to Linear Algebra, I am trying find whether the set given is a Vector Space or not: \begin{equation*} V = \{A\in M_{3\times3} : AA^{t} = -I\}. \end{equation*} I've tried to solve it ...
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1answer
454 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...