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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Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
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Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
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3answers
75 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
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1answer
31 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.
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Determining all scalars $a \in \mathbb{R}$ for which a matrixrepresentation is orthogonal?

Problem: Let $a \in \mathbb{R}$ and \begin{align*} T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}: A \mapsto aA. \end{align*} Determine all $a \in \mathbb{R}$ for which the matrix of ...
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Rule for squaring arbitrary powers?

This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how: $$ (q^{2^{n+1}})^2 = q^{2^{n+2}} $$ I thought it would ...
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59 views

Linear algebra: proving transformation matrix between orthogonal basis is unitary

The vector space $V$ is equipped with a Hermitian scalar product and an orthonormal basis $\{e_1,\ldots,e_n\}$. A second orthonormal basis $\{e_1',\ldots,e_n'\}$ is related to the first one by ...
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21 views

How to multiply the elements within a vector using matrix operations (e.g., dot product)?

Suppose a vector $\vec{v}^T=(v_1, v_2, \ldots, v_n)^T$. To sum the elements within the vector, I can use the dot product with a column vector of ones, $\sum_i v_i = \vec{v}^T \cdot \vec{1}$. My ...
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1answer
44 views

Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$.

Question: Let $V$ be an $n$-dimensional complex vector space, let $T: V \to V$ be a linear transformation. Show that there exists a non-negative integer $r$ s.t. $ker(T^r) = ker(T^{r+1})$. My ...
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42 views

Finding Null Space Basis

let $v$ be a vector $v=(1,-1,1)$, find $Ker(v)$ or $v*x=0$ I have approached it this way $(y-z,-y,z)=(y,-y,0)+(-z,0,z)=y(1,-1,0)+z(-1,0,1)$ But the answer $(1,1,0),(-1,0,1)$ Where am I wrong?
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Show that $T$ is normal

Let inner product space $V$ (finite) above $\mathbb{C}$. Let the operator $T:V\to V$ s.t. $$T^2 = \frac{1}{2}(T+T^*)$$ Prove that $T$ is normal $(T^*T = TT^*)$ $T^2 - T = 0$ So I've tried the ...
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108 views

Consider the ordered basis for the vector space V of lower 2x2 lower triangular matrices with zero trace.

Consider the ordered bases $$ \mathcal{B} =\left\{ \left[ \begin{matrix} -4 & 0\\ 0 & 4 \\ \end{matrix}\right]; \left[ \begin{matrix} 0 & 0\\ ...
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1answer
19 views

What's the difference between these two spaces?

In the finite element method, $Q1$ element is defined by $\textrm{span} \{1, x, y, xy\}$. And $\textit{rotated } Q1$ element is defined by $\textrm{span}\{1, x, y, x^2-y^2\}$. Please tell me what ...
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1answer
42 views

Prove $T|_{V_\lambda}$ is diagonalizable

Let $V$, an $n$-dimensional vector space and let $T, S:V\to V$, two diagonalizable linear operators. Show that if $TS=ST$ then every $V_\lambda$ of $S$ is $T$-invariant and the restriction, ...
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1answer
58 views

Linear transformation: Change of basis

I am given the following linear transformation $L$: $A=\begin{bmatrix}1&2\\0&3\end{bmatrix} \in \Bbb R^{2 \times 2}$ $L: \space \Bbb R^{2 \times 2} \longrightarrow \Bbb R^{2 \times 2}; ...
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Sign of eigenvalues of $A$ by $\det(A-\lambda I)=\lambda \det(B+D-\lambda I).$

Let $A$ be a $n\times n$ matrix, $B$ be a $(n-1)\times (n-1)$ matrix and $D$ be a $(n-1)\times (n-1)$ diagonal matrix with all entries positive. We assume that $$\det(A-\lambda I)=\lambda ...
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1answer
39 views

properties of the solution to a non-homogeneous matrix equation with a non-singular M-matrix

I have a matrix equation $Ax=b$, where $A$ is a $4\times4$ non-singular M-matrix ($A$ has negative off-diagonal and positive diagonal entries) and $b$ is a strictly positive vector. Let $x=(x_1, x_2, ...
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39 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
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50 views

An equality for the dimension of the sum of subspaces (in the non-degenerate case)

This post is a sequel of An inequality for the dimension of the sum of subspaces, inspired by this famous answer on $\dim(U+V+W)$. The inequality $$\dim(\sum_{i = 1}^{n} U_i) \le \sum_{r=1}^{n} ...
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1answer
51 views

Finding the Jordan form and basis failing

Let $$A = \left(\begin{array}{cccc} 3&4&-1\\0&-2&0\\1&-4&1 \end{array}\right)$$ Find the Jordan form $J$ and $P$ such that $P^{-1}AP = J$. So here's what I did: $f_A(x) = ...
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1answer
93 views

Are $A$ and $A^\top$ similar? [duplicate]

Let $K$ be a field and $A$ a square matrix with entries in $K$. Then A and $A^\top$ have the same characteristic polynomial. What do we know about similarity? Do you have an example where $A$ and ...
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42 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
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66 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
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1answer
36 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
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2answers
49 views

How do I solve a linear system with two variables and three equations?

To be specific here is the system: $$x-2y=0 \tag{1}$$ $$x-2(k+2)y=0 \tag{2}$$ $$x-(k+3)y=-k \tag{3}$$ I have already solved it for equations $(1)$ and $(2)$... what should I do with the 3rd ...
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Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
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1answer
72 views

Adjoint and Adjugate are same or different?

The notions of adjoint and adjugate, which I saw, are as follows: (1) Let $T:V\rightarrow W$ be a linear map. Then there is a corresponding linear map between the duals of these spaces: ...
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1answer
112 views

Largest eigenvalue of a Hermitian matrix

I have two Toeplitz positive semi-definite Hermitian matrices $\mathbf{R}_1, \mathbf{R}_2 \in \mathbb{C}^{M \times M}$. They are in fact covariance matrices satisfing the following conditions: (1) ...
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1answer
40 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
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Is it possible to solve for $m$ in a linear equation without knowing $b$?

Suppose you know certain points on a line say $(5,2)$ up to $(8,10)$ but you don't know exactly where the $y$ intercept would be being somewhere down there at like $-25$ area. How would you solve for ...
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4answers
955 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
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361 views

Prove that similar matrices have the same nullity.

How do I approach this? I'm assuming it might have something to do with $B = P^{-1}AP$.
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2answers
78 views

Finding Eigenvalue det(λI - A);

I want to know if what I'm doing to derive equation (2) from (M2) is correct or not; usually, before moving onto the next row in Guass-Jordan elimination we turn a_11 into a leading one or whatever ...
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1answer
34 views

Problem about dual of $W = V \oplus V'$

Let $V$ by finite dimensional, let $W = V \oplus V'$, and prove that the correspondence $(x,y) \rightarrow (y,x)$ is an isomorphism between $W$ and $W'$. (The direct sum is defined as the set of ...
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2answers
41 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
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2answers
61 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
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3answers
24 views

Finding Rank And Eignvalues Of Vectors Multiplication

Let $v=(3,1,3,-4)$ and $A=v^tv$, Find: the rank of $A$ $Null(A)$ eigenvectors and eigenvalues Is there a way to approach this without finding $A$ explicitly?
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1answer
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Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
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599 views

Distributive property of scalar multiplication over scalar addition

I need help with a simple proof for the distributive property of scalar multiplication over scalar addition. Help with proving this definition: $(r + s) X = rX + rY$ I have to prove the truth of the ...
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1answer
73 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
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1answer
85 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
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2answers
48 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
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3answers
67 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
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3answers
40 views

Eigenspace and $\ker(T)$

It seems like eigenspace and $\ker(T)$ are strongly connected, I have thought about some properties and I would like to make sure I got it right. for all matrix/transformation there is an Eigenspace ...
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reoder basis vectors to get 'more diagonal' representation of NxM matrix

I am trying to reorder the basis vectors of an NxM matrix in a way that leads to a representation with 'as much weight close to the diagonal as possible'. I would be happy, if instead of a matrix ...
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2answers
52 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
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2answers
63 views

Every vector space is isomorphic to the set of all finitely nonzero functions on some set

I am trying to prove the statement in the title, that Every vector space is isomorphic to the set of all finitely nonzero functions on some set. A finitely nonzero function from $X \rightarrow ...
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1answer
51 views

Using the Gram -schmidt procedure to find the orthonormal set (Linear Algebra)

(a) Construct an orthonormal basis of the space $R^3$ satisfying the requirment of the Gram-Schmidt prodcure from the basis $v_{1}=(-3,4,0)$ , $v_{2}=(5,10,-24)$ , $v_{3}=(0,0,1)$ (b) Given that ...
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88 views

Possible to express the diagonal matrix $D=\tiny\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right)$ as function of $3\times3$ identity matrix?

Is possible to express the diagonal matrix $$D=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right)$$ as function of $3\times3$ identity matrix? For ...
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2answers
34 views

Size of a triangle using determinant [duplicate]

find the size of a triangle using (determinant) with the following points: $(x_1,y_1)=(1,-2)$ $(x_2,y_2)=(-4,-2)$ $(x_3,y_3)=(-5,-1)$ How should I place those points in the ...