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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Find the projection of a vector onto a subspace of $R^4$

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

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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Any program that turns vectors to orthognal?

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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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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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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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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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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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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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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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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1answer
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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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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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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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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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2answers
19 views

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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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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17 views

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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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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34 views

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
21 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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28 views

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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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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35 views

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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28 views

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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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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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
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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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1answer
14 views

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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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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2answers
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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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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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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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1answer
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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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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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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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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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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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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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I have to show $e_i \in r(T)$ for all $i\neq r$, $1\leq i \leq {n}$.

$D$ be a division ring and $n>2$ a natural number. $e_i$ denotes the element in $D^n$ whose$ (i,j)$-entires are zero. Let $T\in M_{(n-1)\times n}(D)$ such that $T=\begin{pmatrix} T_1 \\ T_2\\ ...
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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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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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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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1answer
19 views

What is 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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19 views

Is it true that $\tilde{P} = D^{\dagger} P D$ has non-negative entries?

Consider a $n \times n$ stochastic matrix $P$ (i.e. non-negative rows sum to one). We are interested in the matrix $\tilde{P} = D^{\dagger} P D$, where $D$ is a $n \times k$ matrix which is ...
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How to express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors?

$u,v$ are linearly independent and $w$ is a non-zero vector. Let $Angle(u,v)=\phi$ and $Angle(u \times v,w)=\theta$. Express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors. ...
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1answer
17 views

Which of the following is true for the following linear transformations?

If $T_1$ and $T_2$ are linear transformation on $V_2(\Bbb R)$ by $T_1(a,b)=(0,a)$ and $T_2(a,b)=(a,0)$ , then which of the following is true 1) $T_1T_2=0$ 2)$T_1^2=T_1$ 3)$T_2^2=T_1$ 4)$T_1T_2 $ ...