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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Transcendental Basis

Can you say that because $\pi$ is transcendental, that a basis of $\{\pi, \pi^2, \pi^3, \dots\}$ in the rational numbers $\mathbb{Q}$ spans the entire real numbers? It seems likely, although I can't ...
4
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4answers
107 views

Prove that for vectors $v_1,…,v_n$ in $\mathbb C^n$, $\{v_1,…,v_n\}$ is a basis for $\mathbb C^n$ iff its conjugate is a basis for $\mathbb C^n$

Prove that for vectors $v_1,...,v_n$ in $\mathbb C^n$, $\{v_1,...,v_n\}$ is a basis for $\mathbb C^n$ if and only if $\{\bar v_1,..., \bar v_n\}$ is a basis for $\mathbb C^n$. I know intuitively that ...
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1answer
70 views

Equivalence relation $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$

Let $(G, \cdot)$ be a group with an Identity element $e$. (i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and ...
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0answers
870 views

Linear algebra Pruning X, linear combinations and Spans

Consider the following subset of the vector space $\mathbb{P}_4(\mathbb{R})$ (real polynomial functions of degree at most 4): $X := \{f_1,f_2,f_3,f_4,f_5 \}$ with $f_1(x) = 1 + x^3 + x^4$, ...
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1answer
82 views

Orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$

Let $A$ be a matrix. The orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$ is unique if and only if the columns of $A$ are linearly independent. True or False?
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1answer
55 views

Unit circle - how to prevent backward rotation

Let's assume we have a unit circle (0, 2$\pi$). Basically I have a point on this circle who is supposed to move only forward. This point is controlled by the user mouse and constantly calculate 25 ...
0
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1answer
259 views

Proving a subspace under a linear transformation by the closure of standard addition and scalar multiplication

$T(x,y,z)= (3x-2y, -2x+3y, 5z)$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ Show that $A= \{(u,v,z) \in \mathbb{R}^3~|~(u,v,w)=T(x,y,z)\}$ for some $(x,y,z)$ in $\mathbb{R}^3$ is ...
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1answer
940 views

Orthonormal Basis for Hilbert Spaces

The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an ...
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395 views

Dimension formula proof

I got stuck understanding the proof from here (page 1, last line). Why is $z \in X \cap Y$? Thanks.
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1answer
283 views

A Book for Linear Algebra [duplicate]

I want to start learning Linear Algebra, I have no background about this subject except high school mathematics that doesn't includes complex number and matrices. I found the following books: ...
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31 views

Linear combination on $\mathbb R_{1} \left[ x \right ]$

Given the canonical basis of $\mathbb R_{1} [ x ]$, $\{x,1\}$, I tryed to write each vector as a linear combination of the basis $B=\{3x+6,2x+10\}$. So, I started by $x$. ...
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2answers
259 views

Einstein Notation for product of stacked matrices

Background Information: I recently started using the Einstein summation notation to express certain operations over an "image" $\mathbf{A}$ where to each pixel a square matrix is attached. That is, ...
4
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1answer
73 views

Is anybody here able to construct this example?

I need to construct an example for such a situation: Let $x_1,x_2$ and $v_1,v_2$ be four vectors in $\mathbb{C}^2$, so that they are mutually different from each other. Further, there have to be ...
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1answer
2k views

Subspace test on polynomial function of degree at most 2

For each of the following, either use the subspace test to show that the given subset, $W$, is a subspace of $V$ , or explain why the given subset is not a subspace of $V$. (c) $V = P_2(\mathbb{R})$ ...
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1answer
51 views

Prove that the Gelfand transform $\widehat{f}$ is uniform algebra

I'm going to find an example of uniform algebra and show that satisfying the definition. Example: Show that The Gelfand transform $\widehat{f}$ is uniform algebra. We know that: A uniform ...
4
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1answer
597 views

The Hodge $*$-operator and the wedge product

On every Riemannian manifold $M$, we can consider the Hodge $*$-operator, which is characterised by the following formula: $$a \wedge *b = (a,b)\nu.$$ Here $a$ and $b$ are smooth forms on $M$, $(\ ,\ ...
4
votes
1answer
30 views

Decompose symmetric matrix to scaling factors

I have a symmetric square matrix $P$ composed by left- and right-multiplying another symmetric square matrix $Z$ with a diagonal matrix $Λ$: $$P = ΛZΛ$$ i.e. ($λ_i$ means $λ_{ii}$): $$ ...
4
votes
2answers
48 views

Powers of matrices equality

let $A$ be a $3$ by $3$ matrix with two eigenvalues $\lambda _1, \lambda _2$ such that $\lambda _1$ has algebraic multiplicity $2$ and $\lambda _2$ has multiplicity $1$. I want to prove that ...
3
votes
1answer
111 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
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1answer
163 views

Factor out a matrix if it's being multiplied by a vector?

I'm pretty sure we can do the following, but I just want to confirm. A$\vec x_1$ + A$\vec x_2$ + A$\vec x_3$ + A$\vec x_4$ = A( $\vec x_1$+ $\vec x_2$+ $\vec x_3$+ $\vec x_4$)
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40 views

Prove $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$ well-defined and linear

Let $V$ and $W$ be finite dimensional vector spaces, and $f : V \to W$ a homomorphism. Show that $\Lambda^kf:\Lambda^kW^* \to \Lambda^kV^*$, defined by $\Lambda^k\alpha(v_1, ..., v_k)=\alpha(f(v_1), ...
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1answer
59 views

Faster Methods to Determine Linear Independence [Poole P474, P476 6.2.50]

From Ex 6.43 of P474 asks ... we check that $\{1 + x, 1 - x, x^2\}$ is linearly independent. Can you see a quick way to tell this? I thought to consider a harder question instead: P476 ...
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2answers
813 views

If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
5
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2answers
108 views

How Find $\cos{(\pi A)}$ if $A$ is Orthogonal matrix

let $A_{n\times n}$ is Orthogonal matrix, Find the value $$\cos{(\pi A)}=?$$ and before I guess $$\cos{(\pi A)}=E-2A$$ Now this is wrong,and this problem relsut is what? I know this ...
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0answers
43 views

What is the difference between the concepts of mutual orthogonality and pairwise orthogonality

Let $u,v,w\in\mathbb R^3.$ What is the difference between the concepts of mutual orthogonality and pairwise orthogonality of $u,v,w\in\mathbb R^3?$
4
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1answer
81 views

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$ There's a similar question floating around but I was merely wondering if the result holds in the same way when we let A and B be fields, ...
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0answers
126 views

some corollaries of the rank - nullity theorem

Here is a problem which I encountered in linear algebra. I realized that it might be a corollary of the "rank - nullity theorem" but I don't know how to work with it. Hope you can help! Thank you! ...
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80 views

Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
2
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1answer
364 views

Symmetric solutions of matrix equation $X-AXB=C$

Consider the matrix equation for unknown matrix $X$ $X-AXB=C$, where $A,B,C$ are symmetric $n\times n$ matrices (can be definite, semidefinite, or indefinite). It is well known that a solution $X$ ...
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1answer
82 views

$V = X \oplus Y$. Will dim$V$ = dim$X$ + dim$Y$ Hold If $V$ Is Infinitely-Dim?

I don't understand much about the infinite-dimensional vector space. Will the statement $V = X \oplus Y \rightarrow$ dim$V$ = dim$X$ + dim$Y$ hold in a infinitely-dimensional case? Let $V$ be ...
2
votes
1answer
269 views

Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question: Are my answers correct? a) $P_{2}$: $ ...
2
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0answers
41 views

Show $U_A$ is a unitary operator.

Let $A$ be an invertible $n \times n$ matrix with real entries. Show that $(U_A f)(x) = f(A^{-1}x) \vert \det(A) \vert^{-1/2}$ defines a unitary operator in $L^2(\mathbb{R}^n,d\lambda)$. I have some ...
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0answers
39 views

For every invertible matrix $S$ there exist bases $B$, $B'$ such that $S = M_{B'}^{B}$

I want to prove: $V$ vector space, $\dim V < \infty$. For every invertible matrix $S=[s_{ij}]$ there exist bases $B$, $B'$ such that $S = M_{B'}^{B}$ So, let $B' = \{b_1', \dots ,b_n'\}$ be a ...
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3answers
41 views

Decomposition of vector space - two linear mappings

Theorem: Given two linear mappings $f,g: V \rightarrow V$ with $f\circ g = g\circ f = 0$ $f+g=\operatorname{id}_V$ $f\circ f = f$ $g\circ g=g$ Then we have $$V=\operatorname{im}(\,f)\oplus ...
4
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1answer
151 views

If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
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26 views

System of linear equations with modulo elements

Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
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2answers
110 views

Why invariance to change of basis is so important in linear algebra?

I'm reading a book on linear algebra and I see that for every new presented concept (from simple vectors and linear functions and up to tensors) we immediately study how does it behave under a change ...
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2answers
346 views

If $U$ is a subspace of a vector space $V$, then it is also a vector space?

If we define $U$ as a subspace of a vector space $V$ then does that mean that $U$ is also a vector space?
2
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1answer
50 views

If $A=LL^T$, is $A\otimes I_3 = (L \otimes I_3)(L \otimes I_3)^T$?

$A$ is a symmetric positive definite matrix and $LL^T$ its Cholesky factorization. $A \otimes I_3$ is the Kronecker product of $A$ with the 3x3 identity matrix. Is the relation $A\otimes I_3 = (L ...
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4answers
85 views

What is the basis of basis?

They say (here, for instance) that you can represent a vector, $\vec v$ as coordinate vector, $[v]_B$, in base, $B$, $$\vec v = v_1 \vec b_1 + v_2 \vec b_2 + \cdots = \begin{bmatrix}\vec b_1 & ...
5
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1answer
120 views

Is this function surjective; infinite linear combination.

Let $A(t):[0,T] \rightarrow \mathbb{R^{n\times m}}$ be continuous function. Let $$U = \text{span}\left( \bigcup_{t\in [0,T]} \text{span}(A(t)) \right)$$ Define function $f(u): L^\infty ...
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0answers
63 views

Nondegenerate bilinear map

Let $V$ be the vector space consists of all $n\times n$ real matrices, and $f$ is a nonvanishing linear function on $V$ such that $$f(AB)=f(BA),\ \forall\ A,B\in V.$$ Show that $g(A,B)=f(AB)$ is a ...
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1answer
162 views

Functions $ \cos(2x)$, $\sin(2ax)$, $1$ independent and dependent

For which value(s) of $a$ are the functions $\cos(2x)$, $\sin(2ax)$, $1$ independent over the real numbers? For which $a$ are they dependent? I thought maybe to equate each (with the use of ...
2
votes
2answers
92 views

What is the maximum number of distinct roots does the characteristic polynomial have?

Let $A$ be a $3\times 3$ matrix with real entries which commutes with all $3\times 3$ matrices with real entries. What is the maximum number of distinct roots that the characteristic polynomial of $A$ ...
2
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1answer
118 views

Rotation/Inner Product between eigenspaces of Hermitian matrices

Given two Hermitian, positive definite matrices $\mathbf{T_1}$ and $\mathbf{T_2}$, is it generally true that the matrix $\mathbf{G}_{ij} = \lvert\langle \mathbf{u}_i \mathbf{v}_j \rangle\rvert^2$ of ...
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1answer
50 views

Find the standard matrix of $T$ with respect to$S=\left \{ 1,x \right \}$ and $S'=\left \{ 1,x,x^{2} \right \}$

$B =\left \{1+x,3+2x \right \}B'=\left \{ 2,3-x,5+x^{2} \right \}$ you are given the matrix of a linear transformation $T:\mathbb{P}^{1}\rightarrow \mathbb{P}^{2}$ with respect to $B$ and $B'$ is: ...
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2answers
142 views

show that the characteristic polynomial of this matrix has negative coefficients

Let $n\geq 2$, $A$ be the $n\times n$ matrix $A=(a_{ij})$ where $a_{ij}=\max(i,j)$. Can anybody show that the characteristic polynomial $P(x)=\det(xI-A)$ has all its coefficients negative except the ...
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2answers
178 views

Find a matrix that has its nullspace spanned by the vectors (1,1,1) and (2,1,0).

I'm no expert at linear agebra, but I have a feeling I made a paper with an impossible question. The question, as stated in the title is: $V$ is a vector space spanned by the vectors ...
0
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1answer
315 views

Find an orthonormal basis for W and $W ^{\perp}$

$V=\mathbb{P}^{2}$ with the inner product $<p(x),q(x)>=2p(-1)q(-1)+3p(1)q(1)+p(2)q(2)$ Let $W=Span${$x,x^{2}$} Find the orthonormal basis for W using Gramm-Schmidt. Then express the ...
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4answers
105 views

Given the set of eigenvalues of a diagonalizable matrix, show that it satisfies an equation

Let $A$ be an $n \times n$ diagonalizable matrix. $A$ has only $2$ and $4$ as its eigenvalues. Show that $A^2 = 6A − 8I$. I get stuck on this question for a while. Can anyone give me a hint for this ...