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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Diagonalization of a Matrix in terms of other matrices and eigenvalue

Task: Let A be a symmetric matrix having only one eigenvalue λ and C be a matrix that diagonalizes A by a similarity transformation. Find a simplified expression for A in terms of λ, C, and I, the ...
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43 views

Monic generator of an ideal

I'm looking for the monic generator of an ideal. In particular, I've already shown that $$M=\{f\in \mathbb{Q}[x]|f(0)=0\}$$ is an ideal because if we take an element from $M$, and multiply it by an ...
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72 views

Invertible, Positive and Isometry Operator.

Let $T ∈ L(V )$ and $T = SP$, where $S$ is an isometry and $P$ is a positive operator. Prove that $T$ is invertible if and only if $P$ is invertible. Here is my approach: $\implies:$ $T = SP$ by ...
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An irreducible character of degree prime affords a faithful representation.

This is a long question, but hopefully someone can give me a suggestion, as I've been hitting my head against the wall... Take a non-abelian group $P$, of order $p^3$, $p$ prime. We've already ...
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Find bases for eigenspaces of A

$$A = \begin{pmatrix} 6 & 4 \\ -3 & -1\end{pmatrix}$$ Find the bases for eigenspaces $E_{\lambda_1}$ and $E_{\lambda_2}$ of $A$. I don't really know where to start on this problem.
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Do elementary row operations give a similar matrix transformation?

So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are ...
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26 views

How do I find the intersection of two vector spaces?

What is the general method to find the basis of the intersection of two vector spaces? Example: $$U= <(1,2,0,0),(0,-1,1,0)>; V= <(1,0,0,-1),(0,1,0,0),(0,0,1,0)>$$
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Show that a matrix is not diagonalizable over a finite field

I have a matrix: $$\ \left[ {\begin{array}{cc} 2 & 2 \\ 1 & 2 \\ \end{array} } \right] $$ which I need to show that it cannot be diagonalized over the finite field $\mathbb{F_3}$. ...
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Show that a linear map $f=\lambda Id$

Let $V$ be finite dim $K-$vector space. If w.r.t. any basis of $V$, the matrix of $f$ is a diagonal matrix, then I need to show that $f=\lambda Id$ for some $\lambda\in K$. I am trying a simple ...
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The definition of $\oplus$

I would like to understand why the books give two different concepts to $\oplus$ between vector spaces: See: Concept 1: $W=V_1\oplus V_2=\{(v_1,v_2)|v_1\in V_1, v_2\in V_2\}$. Concept 2: ...
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Orthogonally diagonalizing a matrix given a linear mapping, without calculating the original matrix

Let f:R^3 ---> R^3 be the linear mapping which reflects over the plane 3x-2y+z=0. The goal of this question is to orthogonally diagonalize the matrix [f] without calculating [f]. (a) Determine an ...
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Verification of an R-Module isomorphism between $R^n$ and its dual

With one step at a time, I am getting slightly more used to $R$-Modules. Let $R$ denote a commutative Ring with $\mathbb{1}$ and $n$ a natural number. For the tuple $a:= (a_i)_{i=1}^n \in R^n$ we ...
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32 views

Change of Basis for a transformation matrix

Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 by $ $$T\left(\left[ \begin{matrix} x_1 \\ x_2 \\ \end{matrix}\right]\right) = \left[ \begin{matrix} x_2 \\ x_1 \\ \end{matrix}\right]$$ Let ...
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21 views

$M_n$ equivalent to the $C^*$-algebra generated by Jordan Block?

Suppose $J$ is the $n\times n$ Jordan Block matrix with all zero eigenvalues, (a.k.a. the "shift" matrix) $\begin{pmatrix} ...
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21 views

equivalnce of linear functions, which one's kernel includes the other's

The following is from my homework. PLEASE don't reveal all the solution, but leave at least something for my imagination. Let $X$ be a normed space. Let $\phi,\psi : X → \mathbb C$ be linear ...
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Prove that if $\ker(T) \subseteq \ker(S)$, then $S = kT$ for some $k\in \mathbb{R}$

Suppose $V$ is a finite dimension linear space with dimension $n$, and that $S,T: V\rightarrow \mathbb{R}$ are linear transformations such that $\ker(T) \subseteq \ker(S)$. What are the ...
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Proof of Linear transformation

Let $T$ be a linear transformation from $\mathbb{R}^n$ to itself. For a given vector $v$ of $\mathbb{R}^n$, if $T(v)\neq 0$ but $T^2(v) = T(T(v)) = 0$, then prove that $v$ and $T(v)$ are linearly ...
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21 views

World Problem on writing an equation

Question: A community centre offers pottery classes. A \$40 enrolment fee covers supplies and materials, including one bag of clay. Extra bags of clay cost \$15 each. Write an equation to represent ...
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Linear Transformation applied to the some of the basis of P3

Let $T: P_3 \rightarrow P_3$ be a linear transformation such that $T(-2x^2) = (-3x^2 -4x)\\ T(-0.5x+3) = (-2x^2 -2x -4)\\ T(5x^2 +1) = (-2x-4).\\$ Find $T(1), T(x), T(x^2) and T(ax^2+bx+c)$ where ...
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45 views

Diagonalizing zero matrix

Consider the matrix $A = 0$ that is diagonalized by the matrix $$S = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}.$$ What is the diagonal matrix? I'm confused because I thought you could ...
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$A y= b$ in $C(X)$

Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of all complex valued continuous functions on $X$. If $A\in C(X)^{m\times n}$, $b\in C(X)^{m\times 1}$, and for all $x\in ...
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31 views

Linear Algebra Spanning question

If I have two 3x1 column vectors in a vector space V that are linearly independent, how can I make a system of 3 eqns whose solution will span V? For example, column vector [1,3,0], column vector ...
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Can I decompose an arbitary matrix as the sum of antisymmetric, identity, and outer product parts?

I'm trying to see if I could represent any arbitrary $3\times 3$ matrix by the following matrix: $$\begin{bmatrix}0 & -c & b \\ c & 0 & -a \\ -b & a & ...
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Projection of n-Simplex into k-Simplex

I try to find properties of orthogonal projections such that a standard n-Simplex $S_n$ is projected into a k-Simplex $S_k (k\leq n)$. Literature provides work on "smallest projections" in this ...
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Have I understood linear transformations correctly?

I've just started self-studying Linear Algebra, and I am not quite sure I completely understand linear transformations. Here' what I've gathered for so far... Given a matrix $A$, we can define a ...
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In what sense is the Bayesian posterior mean a “convex combination”?

This is related to a previous question that hasn't gotten an answer: Definition of convex combination with matrix-vector multiplication Suppose I want to estimate $x \in \mathbb{R}^n$ from two ...
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43 views

Derivative and Antiderivative operators in Hoffman Kunze

I have a homework problem in Hoffman-Kunze; Let F be a subfield of $\mathbb{C}$ and let $T,D$ be the transformations of F$[x]$ defined by ...
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24 views

Prove that Least squares and decorrelator are equivalent

Here is the problem: $$\mathbf{y}=\mathbf{Ax}+\mathbf{b}$$ where $\mathbf{y,x,b}$ are vectors, and$\mathbf{A}$ is matrix(generally rectangular, but with full column rank). The least squre solution ...
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Converting normalised values into original

I have a normalisation formula as follows, which takes a list of numbers, such as $1,2,3,4,5,6,7,8,9,10$, and returns the normalized values which guarantees that $\tilde{x_i} \in [0,1]$. ...
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Show tha triadiagonal is M-matrix.

How to show that a tridiagonal matrix $A=(-1,2,-1)$ is an M-matrix, meaning that the entries of its inverse are nonnegative?
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41 views

Eigenvalues, polynomials and minimal polynomials

I have proved (a) by: Let $\lambda$ be an eigenvalue of $AB$ $ABv=\lambda*v$ Then $BABv=\lambda*B*v$ so Bv is an eigenvector of BA with eigenvalue $\lambda$. For B, I have found the formula in ...
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Diagonalisable matrices over different fields

I believe this fits in with my knowledge of Jordan Normal form, however I am not sure how to approach the question itself? I am especially lost with $F_7$
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The design matrix X for this linear model?

Hi thanks for the feedback. The full question is: Consider a randomised block design consisting of 3 blocks ($B_1$, $B_2$, $B_3$) and each block consisting of 3 plots. Furthermore there are three ...
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Why if we have representation $\rho$ of finite group then $\rho(g)$ is diagonalisable matrix?

Why if we have representation $\rho:G \to GL(V)$ of finite group $G$ then $\rho(g)$ is diagonalisable matrix? I read that it's because $x^{o(g)} -1$ splits, but I don't understand how this fact is ...
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65 views

Compute the dimension of a 5x5 matrix

The matrix is given like this: all A(ii) entry= 1/2 while A(ij) entry= -A(ji) entry. (In other words, all the diagonal entries of this 5x5 matrix is 1/2 and all the off-diagonals is A(ij) entry=-A(ji) ...
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256 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
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How to find a matrix $Y$ with $Y^2≠0$ while $Y^3=0$. [closed]

I'm asked to find a $3\times3$ matrix $Y$ where $Y^2≠0$ while $Y^3=0$. May I ask is there any method to solve such a question? thank you!
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What is the best explanation to say these vectors are linearly dependent?

(2.7,18,28),(57,7.2,15),(3.14,159,2.6),(161,803,3.9) < 4vectors.Justify whether they are linearly indepedent or dependent. My idea is linearly dependent because each vector cannot be expressed as ...
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55 views

Commuting matrices implies upper triangular simultaneously

Let $A_\alpha$ be a family of commuting matrices, that is, $A_\alpha A_\beta=A_\beta A_\alpha$. Show that there exists an unitary matrix $U$ such that $U^*A_\alpha U$ is upper triangular for each ...
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Prove that if $T^3$ = $T^4$, then $T^2 = T$ and there exists a polynomial $f(x) \in P(C)$ such that $T^* = f(T)$

Let V be a finite-dimensional complex inner product space and $T \in L(V)$ a normal operator. Prove that (i) If $T^3$ = $T^4$, then $T^2 = T$ (ii)There exists a polynomial $f(x) \in P(C)$ such that ...
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Dimension of a subspace of $M_n(\mathbb C)$. [closed]

What is the dimension of the subspace of $M_n(\mathbb C)$ consisting of matrices $B$ such that $AB = BA$, for some fixed matrix $A$ in $M_n(\mathbb C)$?
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Linear transformation with clockwise rotation on z axis

Let $T$ be a linear Transformation from $\mathbb{R}^3$ to itself such that $T$ is $60^{\circ}$ clockwise rotation with fixed $z$-axis (i.e, rotate the space according to the $z$-axis) where ...
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Matrix with linear transformation with reflection

Find the matrix of the linear transformation A which is the reflection in the line $y = \sqrt{2}x$ with respect to the standard basis in $\mathbb{R^2}$. I Have no idea how to approach this problem... ...
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finding dimension of subspace of $P_3$ given by $H=\{a+bx^3:a,b \in \mathbb R\}$

I'm not sure how to find the dimension of this set or any set like this based on what I know about dimensions. $H=\{a+bx^3:a,b \in \mathbb R\}$ All it seems I'm given in my notes is that $\dim(P_3) ...
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Linear differential equation and harmonic motion problem

An atom undergoes simple harmonic motion. Initially its displacement is $1$, its velocity is $1$ and acceleration is $-12$ compute its displacement and acceleration when the velocity is square root ...
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Find the least square solution and optimal least square solution of the linear system

Find the least square solution and optimal least square solution of the linear system: $$x_1 + 2x_3 = 1$$ $$x_2 +3x_3 = 0$$ $$-x_1 + x_2 + x_3 =0$$ $$-x_2 -3x_3 =1$$ Letting $A$ be the matrix of ...
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What is an elementary yet important application of matrix in finance?

What is an elementary yet important application of matrix in finance? I have difficulty to read anything intermediate/advanced associated with this topics, hopefully I can find something interesting ...
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Eigenvalues and eigenvectors of similar matrices.

Suppose there is a transformation $T$ and let $A$ be a matrix representation of $T$ with chosen basis. If I find out the eigenvalues of matrix $A$, these eigenvalues will be the eigenvalues of the ...
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74 views

Find a three independent vectors u, v, w that each lie in N(A), the null space of A.

$A=\begin{pmatrix} 2& 4& -4& 4\\ -1& -2& 2& -2 \end{pmatrix}$ Find a three independent vectors $u, v, w$ that each lie in $N(A)$, the null space of A. Can someone ...
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Proof of inequality for singular values

I have two questions: I wonder how to prove inequalities involving the singular values of matrix A*B such as Using the definition meaning that largest singular value is max ||AX|| over all ||X||=1 ...