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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$\dim (W_{1} \cap W_{2}) = \dim W_{1}$ implies $W_{1} \subset W_{2}$?

Let $V$ be a finite-dimensional vector space and let $W_{1}, W_{2}$ be subspaces of $V$. If $\dim (W_{1} \cap W_{2}) = \dim W_{1}$, must $W_{1} \subset W_{2}$? Since $\dim (W_{1} \cap W_{2}) = \dim ...
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0answers
14 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
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0answers
4 views

generalized inverse and its applications

As generalized inverse has vast applications in the field of linear algebra, but why the generalized inverses is important? why we are studying about it?
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0answers
14 views

tensor of two vector space

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
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1answer
17 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
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1answer
9 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
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1answer
35 views

An example of non euclidean inner product

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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0answers
18 views

The dual of the dual, isomorphism, equivalence of functions

Problem: Let $V$ be a f.d. vector space. Define $\theta:V\to (V^*)^*$ given by $\theta(v)(\alpha)=\alpha(v).$ Let $T:V\to V$ and $T^*\ ^*:(V^*)^*\to(V^*)^*$ be linear maps. Prove $T=T^*\ ^*$. ...
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1answer
11 views

Basis, polynomial vectors

Given the vector space $P_3(R)$, find a basis for it containing the polynomials $x^2 + 1$ and $x^2 - 1$. To find a basis, I need to find whether there exists constants in front of these two vectors ...
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2answers
26 views

Intuitive explanation why if $P$ is a subspace of linear space $L$, then $L/P$ is not a subspace of $L$

Is there an intuitive explanation of why: if $P$ is a subsppace of linear space $L$, then $L/P$ is not a subspace of $L$. I know that it is true, but it is counter intuitive to me.
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0answers
25 views

Which space it belongs to Hilbert, Banach or something else?

The question is related to the following two questions. The link: Understanding Eigenvector defines the problem at hand. The question is the in which space (Hilbert or Banach or something else) we ...
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0answers
12 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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0answers
12 views

Induced map between quotient vector spaces well defined and linear?

Suppose we have vector spaces $V,W$ and subspaces $$ V\supset V'\supset V''\\ W\supset W'\supset W'' $$ suppose also that we have a linear map $A:V\to W$. What does it take for this map to induce a ...
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2answers
24 views

Is $(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$?

Is this true? $$(A \oplus B)^{\perp} = A^{\perp}\cap B^{\perp}$$ I am trying to prove this, but could not find a way. Any suggestions would be much appreciated. Thanks.
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1answer
34 views

Eigenvectors of derivative

I'm trying to consider how linear algebra relates to calculus. It seems to me that the only eigenvectors of the derivative operator on $\Bbb R$ are the functions $ce^{kx}$ for constants $c$ and $k$. ...
1
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1answer
17 views

Why do the 1's in Gauss Jordan RREF need to be along main diagonal and not other diagonal?

I've practiced G-J elimination and understand most of the algorithm insofar as it represents the different manipulations one can apply to a system of equations. However, when we're talking about ...
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1answer
21 views

Linear Algebra-Vector Subspaces Question [on hold]

Let $U=\{f\in C^1([-1,1],\Bbb R);y'=y+1\}$. Is $U$ a subspace of $C^1([-1,1],\Bbb R)$?
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0answers
27 views

Rates and Linear Equations [on hold]

The following question I found in an old high school textbook I bought in a second hand bookshop. The question is exactly as it appears in the text with no additional information. The answer, with no ...
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1answer
24 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
1
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0answers
11 views

Minimal column sum (?) solution to system of linear equations over $\mathbb{Z}_2$

There are $n$ equations in $n$ unknowns where both the coefficients and unknowns come from the field $\mathbb{Z}_2$. I can represent these as the equation $Ax = b$ where $A$ is an $n\times n$ matrix ...
4
votes
2answers
77 views

What space to use?

My apology if this question is not mathematical. I have heard of many spaces, Hilbert space, Banach space etc. But could not connect a specific problem to a space. For example if I ask a mathematical ...
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2answers
28 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
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2answers
34 views

Is the sum of two projections a projection?

Let $ S $ and $ T $ be two linear subspaces of $ \Bbb{R}^{2} $. Then is the sum of the projections $ P_{S} $ and $ P_{T} $ (i.e., $ P_{S} + P_{T} $) a projection? I don’t think it is since the ...
2
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0answers
13 views

Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
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1answer
20 views

Equation for the curve in terms of x,y

we got the equation $$r(t) = (t-2)i + (t^2+4)j$$ I got $$x = 1-2t$$ $$y = 1+4t$$ Would that be correct?
2
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0answers
48 views

What is the point of basis vectors?

Why do we even bother with basis vectors? Why don't we just notate an element $x$ of an $n$-dimensional vector space $V$ as an ordered set $(x_1,x_2,...,x_n)$ and go from there?
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1answer
28 views

For what values of a constant does the system have: No solution; More than one solution; Unique solution [on hold]

Consider the linear system . For what values of a does the system have: a) No solution; b) More than one solution; c) Unique solution It should be answered by augmented matrix.
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3answers
28 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
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2answers
33 views

For which value of k will the vector be a combination of two vectors?

For which value of $k$ will the vector $\begin{bmatrix}1\\-2\\k\end{bmatrix}$ in $\mathbb{R}^3$ is a linear combination of the vectors $w=\begin{bmatrix}2\\-1\\-5\end{bmatrix}$ and ...
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0answers
8 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
1
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5answers
70 views

Product of any two arbitrary positive definite matrices is positive definite or NOT?

Suppose that , $A$ and $B$ are $n\times n$ positive definite matrices and > $I$ be $n\times n$ identity matrix. Then which of the followings are positive definite ? (i) $A+B$ (ii) $ABA$ ...
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0answers
5 views

Find all vertices of a parallelepiped given only 3 to start with (linear algebra)

My question is simple. I just want to find out the rest of the vertices given only three of them. I haven't really grasped the process of finding the vertices, so I need someone to help me understand ...
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0answers
9 views

Find an orthogonal basis of inner product

Let's define dot procduct $<A,B>=Trace(A B^T)$ over $M_{n \times n}(\mathbb{R})$ Find basis or system of equations describing an orthogonal $W^\perp$ subspace to subspace $W$ which consist of ...
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0answers
17 views

what is number of invertible matrix m*m on$ Z_n$?

‎‎please help me what is number of invertible matrix $‎m*m$‎‎‎ on Group $\mathbb{Z}_n$ ?‎‎‎, assuming we know‎ this number in $\mathbb{Z}_p \quad$ is $‎(p^{n}-1)(p^{n}-p) \cdots ...
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0answers
29 views

Why are Cayley graphs of $SL_2(\mathbb{F}_p)$ defined in this peculiar way?

Naively it would have been natural to pick the symmetric group of generators from members of $SL_2(\mathbb{F}_p)$ but instead one defines a set $S_p$ which is a "natural projection modulo $p$" of an ...
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1answer
13 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
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0answers
22 views

Linear Algebra 3x3 matrix diagonalization Row operation before inversing

Hello I am diagonalizing the matrix $$\begin{bmatrix} -1 & 2 & 2 \\ 2 & 2 & -1 \\ 2 & -1 & 2 \end{bmatrix}.$$ The eigenvalues I found are $-3$ and $3$. The eigenvectors are ...
4
votes
4answers
331 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
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2answers
29 views

simultaneous equation question [on hold]

2x - 3y = 6 || -⅔x + y = 1 what is x and y ?
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0answers
18 views

Overdetermined linear system solutions proof

Let $A\in M_{mxn}(R)$, with $m>n$.Consider that the only solution of the linear homogeneous system $Ax=0_{R^m}$ is the trivial solution $x=0_{R^n}$. Show that linear system $A^ty= b$ have solution ...
1
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2answers
27 views

Eigen space of $T_{A}$ where $T_{A}(v)=Av$

Let $T_{A}:\mathbb{C}^3\rightarrow \mathbb{C}^3$ ,$T_{A}(x,y,z)=A\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ and $A=\begin{pmatrix} 1 & 5 & 0\\ 0 & 1 & 0\\ 0&0&3 \end{pmatrix}$. I ...
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0answers
43 views

Which of these are true? And why? [on hold]

Which of these are true? And why are they true, please answer this too, if possible?
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1answer
11 views

properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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1answer
18 views

(direct) sum vs span of subspaces

Is there a difference between the span of subspaces and the sum of them in linear algebra? They both seem to just be the set of all linear combinations.
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1answer
10 views

Is there any shortcuts in getting an H-infinity norm of a matrix expression?

One of the past exam problems I was solving, has this in its official solution: Usually, to calculate the $H_{\infty}$ norm of any matrix expression $M$ I'd first calculate the eigenvalues of ...
2
votes
1answer
36 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
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0answers
10 views

Show that W is a Subspace of R³

can you help me: Let $u=(1,2,-3)$ and $v=(-2,3,0)$ Two Vectors in R³ and let W the subpace of R³ that consists of all the vectors shape $au+bv$, where, $a,b ∈ R $ show that W is subspace of R³ im ...
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1answer
13 views

Polarization Identity: Sesquilinearity

Problem Given a vector space $V$. Consider quadratic forms with: $$q[u+v]+q[u-v]=2q[u]+2q[v]$$ Then one has a 1-1-correspondence: $$q_s[v]:=s(v,v)\quad ...
0
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0answers
9 views

How to decrypt a ciphertext by using the mutual index of coincidence?

I am trying to decrypt a Vigenére cipher text. I have found the key length by computing Index of Coincidence of substrings. The key length is 12. I know the letter frequencies the string and the ...
0
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2answers
42 views

Propositions of elementary matrix

i'm trying to solve a question about elementary matrix. When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix. Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$; ...