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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0answers
34 views

norm of matrix 1 [on hold]

SHOW ∥A∥1=∥AT∥∞? i dont solve.....
2
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3answers
106 views

How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
0
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1answer
24 views

How to count algebraic multiplicities to show $\nexists$ an eigenbasis for $A$?

If $A=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix},f_A(\lambda)=(1-\lambda)^3 \,\text{and } E_1=\text{ker ...
0
votes
3answers
54 views

For which values of a do the following vectors form a linearly independent set in R^3

I've seen this same question, but asking for linearly dependent, not linearly independent. $$ V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, ...
0
votes
0answers
15 views

Solution space of Linear homogeneous differential equation

The solution space of a L.H.D.E of order n is a vector space spanned by n base vectors, right? So any solution is then a vector of the solution space -> a linear combination of the base vectors. But ...
-1
votes
0answers
34 views

Fill in the missing entries of matrix $Q$ to make it orthogonal [on hold]

I am given the following matrix $Q$: $Q=$ where $p1,p2,...,p8$ are unknowns. I need to make $Q$ into an orthogonal matrix. It occurs to me that $v1 =\{1,1,1,1\}$ and $v2=\{2,1,0,-3\}$, but I'm ...
3
votes
0answers
21 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
2
votes
1answer
31 views

Affine Transformations: Book to Study over the Summer

I've briefly heard of affine transformations in both linear algebra and calculus and I'd like to find a good book on the subject to study over the summer. So what's a good undergrad-level book on ...
-4
votes
1answer
28 views

Orthogonality and inner product [on hold]

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
0
votes
2answers
34 views

Nilpotent matrix similar to a matrix $[0,X]$ where $X$ is full column rank.

I am trying to prove that a nilpotent matrix $N$, which has a Jordan Form consisting only of blocks which are order 2 or greater, is always similar to a matrix $\begin{bmatrix}0 & X\end{bmatrix}$ ...
0
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0answers
22 views

Showing linear functionals are linearly independent

In general: Given $f_1,f_2,...,f_n\in W^*$. To show they are linearly independent, will it be enough to take the standart base of $W$,$B=\{e_1,...,e_n\}$ and its dual base $B^*=\{g_1,g_2,...,g_n\}$ ...
4
votes
3answers
62 views

Where does $\Lambda=P^{-1}AP$ come from?

How do we derive the fact that if a matrix is diagonalizable then we can diagonalize it with the formula $\Lambda = P^{-1}AP$, where $P$ is a block matrix whose columns are the eigenvectors of $A$? I ...
1
vote
1answer
25 views

Rank of a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$ is equal to $m$?

Take a linear map $f: \mathbb{R}^n \to \mathbb{R}^m$, where $n>m$. Is the rank of $f$ always equal to $m$? Since the image of $f$ contains $\{f(a) | a \in \mathbb{R}^n \}$, the image will contain ...
1
vote
1answer
25 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
0
votes
1answer
11 views

Linear dependence under transformation

I have a linear map $f:\mathbb{R}^6\rightarrow\mathbb{R}^4$ and I'm asked to show that if $u$,$v$ and $w$ are linearly dependent vectors in $\mathbb{R}^6$ then also $f(u),f(v),f(w)$ in $\mathbb{R}^4$ ...
0
votes
1answer
21 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
1
vote
1answer
20 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
0
votes
1answer
18 views

Find vector and parametric vector of a line

I have a line that is perpendicular to a plane. This perpendicular line is $3i-2j+6k$. I've also been given that the line passes through $A(2,3,0)$. I'm unsure on how to represent this line as a ...
1
vote
3answers
57 views

How do I find the determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$?

I am getting crazy with these determinants. For a little, I thought I could solve a problem alone, because I had understood more or less how to calculate the determinants of a matrix, but I am back to ...
1
vote
1answer
22 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
-1
votes
1answer
15 views

Null space and Matrix equations

http://studyguide.pk/Past%20Papers/CIE/International%20A%20And%20AS%20Level/9231%20-%20Further%20Mathematics/9231_s03_qp_1.pdf I would like to know the method to answer question 8. I have been having ...
1
vote
1answer
56 views

What is the difference between $A^{-1}$ and $A^\Theta$?

Let $A$ be a square invertible matrix. Then $$A \cdot A^{-1} = I$$. Let $A^\Theta$ be the conjugate transpose matrix of $A$. Then $$A \cdot A^\Theta = I$$. Both on multiplication with $A$ gives ...
1
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0answers
24 views

Proving that a linear functional is matrix trace [duplicate]

Let $W=\operatorname{M}_{n\times n}(\mathbb{F})$ (square matrices $n\times n$ over $\mathbb{F}$), and $f\in W^*$. If $f(AB)=f(BA)$ for every $A,B\in W$ and $f(I)=n$ prove that ...
-1
votes
0answers
9 views

How do I find a matrix for this linear transformation

The problem is that the transformation is defined by T(p)=(p(0), p(1) P(-1), p(0)) B is the standard basis for M22 and B' = {1, x, x^2}. How ...
1
vote
1answer
11 views

How to find linear equation from text

A company produces covers for cell phones. The costs of a production of $x$ units can be described by a linear function $C(x)=ax+b$. At a production of $150$ units the costs are DKK $6000$. At a ...
0
votes
3answers
27 views

Find the vector passing through a given point which is orthogonal to a given triangle in space

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$. I need to find the vector passing through point $A$ that is perpendicular to the triangle made by ...
2
votes
0answers
20 views

Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
0
votes
3answers
31 views

$\dim (W_{1} \cap W_{2}) = \dim W_{1}$ 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 ...
1
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0answers
40 views

Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
0
votes
0answers
20 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 ...
0
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0answers
8 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?
0
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0answers
38 views

tensor of two vector space [on hold]

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 ...
1
vote
1answer
102 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 ...
1
vote
1answer
23 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 ...
1
vote
2answers
51 views

An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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votes
0answers
23 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 $T=T^*\ ^*$. ...
0
votes
1answer
19 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 ...
0
votes
2answers
36 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.
0
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0answers
32 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 ...
0
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0answers
19 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 ...
0
votes
2answers
28 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.
1
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1answer
38 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
vote
1answer
20 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 ...
-1
votes
1answer
22 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 $C^1([-1,1],\Bbb R)$?
0
votes
1answer
79 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
14 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
82 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 ...
0
votes
2answers
34 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 ...
1
vote
2answers
35 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
votes
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 ...