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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Understanding Extension of Scalars in a Vector Space

$\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector ...
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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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Cauchy-Schwarz, how does it work in this example?

Consider $\|\cdot\|_2$ such that $\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}$. Let $A \in \mathbb{R}^{n\times n}, x\in \mathbb{R}^n$, then $$\begin{align} \|Ax\|_2^2 & = \sum_{i=1}^n ...
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Any program that turns vectors to orthogonal? [on hold]

Are there any sites that can transform S(set of vectors) into an orthogonal basis for R^n? I want to know if I did my problem correctly and would like verification. my vector set is [1 ,2, -1][1, 3 ...
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find the rank of a linear mapping such that $T^2=0$

Let $T:\Bbb R^6\to\Bbb R^6$ be a linear mapping such that $T^2=0$.Then which one is true? a)Rank$(T)$ is less than or equal to 3 b)Rank$(T)$ is greater than 3 c)Rank$(T)$ is equal to 5 ...
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relation between trace of product and sum of matrices?

Given A and B positive definite matrices. Is there an inequality relation between trace(AB) and trace(A+B) ?
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889 views

Monotone matrix

A real matrix $A$ is called monotone if $Ax\geq 0$ implies $x \geq 0$. If inverse of $A$ exists and is real, then prove that $A$ is monotone if and only if inverse of $A \geq 0$. ($x\geq 0$ means $x$ ...
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1answer
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Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
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2answers
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$A_{n\times n}$ be a real matrix, $\lambda\in\mathbb{R}$, $(A-\lambda I)^kv=0$

$A_{n\times n}$ be a real matrix, $\lambda\in\mathbb{R}$, for some nonzero vector $v\in\mathbb{R}^n$ we have $(A-\lambda I)^kv=0$ for some $k$ positive integer. Then could anyone tell mew which of ...
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Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$

To find the characteristic polynomial of the matrix $A^2$, would I just compute $$(\lambda^2+4\lambda-5)^2 ?$$
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Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
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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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3answers
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Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$

We want to find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$ for an arbitrary "n". I have tried writing out a few elements of the sequence as $n \to ...
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Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ...
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1answer
23 views

Find the linear transformation that is a reflection through the line $x=y$

Which of the following $2\times 2$ matrices corresponds to a linear transformation that is a reflection through the line $x=y$ in $ \Bbb R^2 $ ? a) $\begin{pmatrix} 1 & 0\\ 0 & -1 \\ ...
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Suppose $AB=BA$ and $A^{1965}=B^{2015}=I$. Prove that $A+B+I $ is invertible.

Supppse $A $ abd $B $ are matrices, $AB=BA $ and $A^{1965}=B^{2015}=I $. Prove that $A+B+I $ is invertible. I want to prove that $(A+B+I)C=I $ I have no idea how to start. Can any one give some hint? ...
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1answer
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Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\v b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
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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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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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320 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
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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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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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2answers
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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
69 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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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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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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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
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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
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$A$ 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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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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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 ...