37
votes
15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
1
vote
2answers
32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
1
vote
4answers
139 views

Suggestion for a book on Linear Algebra [duplicate]

Please suggest a Linear Algebra book with an introduction and rigorous theory (description) on Eigenvectors , eigen-values , Cayley-Hamilton theorem , Diagonalisation of matrices ; Quadratic forms ( ...
1
vote
1answer
37 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
0
votes
0answers
17 views

Volume of parallelepiped gets smaller when using projection vectors

Given a Euclidean Space R and a subspace R' (of dimension $\geq$m), consider vectors $x_1,...,x_m \in$**R**, and let $V[x_1,...,x_m]$ mean the volume of an m-dimensional parallelepiped formed by those ...
1
vote
0answers
34 views

Generator Matrix

I have a C in $F_2^6$ $(x_1,x_2,x_3,x_4) \to (x_1,x_2,x_3,x_4,x_1+x_2,x_3+x_4)$ for $x = (1,0,1,1)$ i get $c = (1,0,1,1,1,0)$ we know that $$c = G . x$$ G is the Generator Matrix in the solution ...
5
votes
3answers
53 views

Area preserving transformation in a higher dimensional space is unitary.

In $\mathbb{R}^3$, a linear operator $Q:\mathbb{R}^3 \to \mathbb{R}^3$ preserves the area of parallelograms: that is, given $x,y\in \mathbb{R}^3$, the area of a parallelogram formed by $x$ and $y$ is ...
0
votes
1answer
34 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
0
votes
0answers
27 views

symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalue

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalues ($\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is an ...
4
votes
4answers
166 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
2
votes
2answers
73 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
2
votes
2answers
39 views

rank of a matrix with two columns s.t. their dot product is zero

I have function $\sigma(u,v)=(f(u,v),g(u,v),h(u,v))$ s.t. $\sigma_u$ x $\sigma_v\neq(0,0,0)$ (cross-product) also, there is the $3\times 2$ matrix : $$ ...
1
vote
2answers
49 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
1
vote
2answers
33 views

Showing that two Matrices are not similar over GL$_n(\mathbb{F}_2)$

Problem: Show that $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ Are not similar over GL$_n ( \mathbb{F}_2)$ Note: We write ...
5
votes
1answer
75 views

$F,G \in \text{End} (V)$ share the same eigenvalues for $F \circ G$ and $G \circ F$

Problem: Let $V$ be a finite dimensional Vector Space over a field $\mathbb{F}$ and $F,G \in \text{End}(V) $ Show that $F \circ G$ and $G \circ F$ have the same Eigenvalues $\lambda$ My ...
3
votes
2answers
74 views

Linear Algebra Self Study

I'm currently a high school student with a love for math. I have taken Plane and Coordinate Geometry, both Algebra I and II, Trigonometry, and am halfway done with Calc A. I want to major in quantum ...
0
votes
1answer
45 views

Pre requisites of linear algebra

I want to learn abstract linear algebra. Do I require the knowledge of discrete mathematics before I start? I have the impression that abstract maths and their proofs can be understood easily by the ...
8
votes
4answers
176 views

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...
0
votes
1answer
18 views

Question regarding dimension of linear transformation.

I saw in an exercise that if $T$ is a linear transformation $T: V\rightarrow W$ and $T_2: W\rightarrow Z$ and $T_1: X\rightarrow V$ are invertible then the rank of the composition doesn't change. So, ...
0
votes
1answer
61 views

Checking if systems of linear equations are equivalent

I'm going through Hoffman and Kunze's Linear Algebra on my own and can't for the life of me figure out part of Exercise 3 of Section 1.2: Are the following two systems of linear equations equivalent? ...
4
votes
1answer
56 views

Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
8
votes
1answer
218 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
1
vote
2answers
208 views

Endomorphisms on finite dimensional vector spaces $f: V \to V$ are surjective $\iff $ injective

Similar questions: Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective Proposition: Let $V$ be a finite dimensional vector space over an arbitrary field $\mathbb{K}$. If $f: ...
1
vote
1answer
66 views

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? What is $\dim(\mathbf{X^TX})$?

Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? Is this true in general, please? And what is $\dim(\mathbf{X^TX})$, please? Does it equal to $\dim(\mathbf{X})$ ...
1
vote
0answers
40 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
0
votes
2answers
74 views

Is there any formula for summation?

$$0.01\sum_{x=1}^{30}(0.99)^{x-1} = 1-0.99^{30}$$ I wonder if there is a formula for summation and I want to know. Would anyone mind telling me? It would be better for me to solve problems, like the ...
0
votes
0answers
164 views

Orthonormal matrix and diagonal matrix.

Consider the matrix $$A=\begin{pmatrix}3&-1&1\\-1&3&-1\\1&-1&3\end{pmatrix}.$$ (a) Verify that $x=[3\,\,4\,\,1]^T$ is an eigenvector of the matrix $A$ and determine the ...
0
votes
1answer
89 views

Linear Algebra: Matrix and determinant

For 1(a), is $p =12$ and $q = 6$? For b(i), is the answer $a=b$ where $a$ and $b$ do not equal to 0? for b(ii), is the answer $a\ne b$? for b(iii), is the answer $a=b=0$ and the solution is ...
-1
votes
4answers
133 views

Linear Algebra and Set Theory book recommendations.

I would like to studying linear algebra and set theory. Does anyone have a a good recommendation of books/resources/etc.?
0
votes
0answers
26 views

Error of the norm of solution in linear least-squares

How can we estimate the solution norm ($\Vert x \Vert$) error, separate from the solution ($x$) error in solving $Ax=y$ (linear least-squares problem)? Is the error of $\Vert x \Vert$ higher or lower ...
1
vote
1answer
48 views

Linear Space and Inner Products

Prove that the set $$ V=\{f\quad|\quad f:\mathbb{R} \to \mathbb{R}, f \quad\text{is absolutely integrable over} \quad \mathbb{R} \} $$ is a linear space over $\mathbb{R}$. Is it necessary to go over ...
0
votes
1answer
257 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
3
votes
1answer
123 views

linear algebra from beginner to intermediate level

I was wondering if there is a way to learn linear algebra from beginner to advanced level by studying it myself. I want to collect a number of books, video lectures, tutorials and other resources to ...
3
votes
3answers
377 views

Proving linear independence of vectors which are functions of other independent vectors

If the $n$-component vectors $a,b,c$ are linearly independent, show that $a+b, b+c, a+c$ are also linearly independent, Is this true of $a-b,b+c,a+c$? What I did was write the new vectors as sums ...
3
votes
1answer
187 views

Constructing a non-linear system with prerequisites about the nature of its critical points.

An exercise from the book I am reading is: "Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus." I have tried many systems. I ...
1
vote
1answer
538 views

How to approach the problems in the Hoffman Kunze book?

I’m considering the idea to solve the entire set of problems contained in the Hoffman-Kunze text. The standard results of Linear Algebra are known to me. I don’t have much time to read the textual ...
6
votes
3answers
275 views

Beginning with math

I am studying computer science since 3 years now. It is really math heavy and I like it. However the problem that I have is that I never really had math in school it was too basic and I lack some ...
3
votes
2answers
91 views

Confusion on Eigenvalues of Similar Matrices

Please help me to identify where I went wrong: The completely reduced normal form of the real matrix $A= $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ ...
2
votes
5answers
134 views

What am I doing wrong in calculating this determinant?

I have matrix: $$ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 3 & 3 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \end{bmatrix} $$ And I want to calculate ...
13
votes
1answer
3k views

How to self study Linear Algebra

I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is ...
7
votes
2answers
2k views

Use of determinants

I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
0
votes
1answer
341 views

Real Understanding definition/theorem, example for Linear Algebra

I have something to discuss about mathematical understanding, and use the Linear Algebra as example. Sometimes, I feel the intuitive understanding of definitions/theorems itself is very different ...
3
votes
1answer
350 views

Notation in linear algebra, what are $N(T)$ and $R(T)$

Working through some stuff I found on the web, I came across a notation that I haven't seen in my textbooks. In this problem, $ T: P_4(\mathbb R)\rightarrow \mathbb R^4 $ is a linear transformation, ...
4
votes
3answers
364 views

First Course in Linear algebra books that start with basic algebra?

I'm 30 years old, and the only math I can remember from college is basic algebra and some probabilities. Next month, I have a machine learning project I'd like to work on, but I'll need a solid ...
5
votes
2answers
4k views

Rigorous Text in Multivariable Calculus and Linear Algebra

So I'm wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus ...
1
vote
1answer
263 views

Frequency on doing exercises in learning Calculus & Linear Algebra & Probability and so on

When I'm taking courses in Calculus I & II, and Linear Algebra, the lecturers are always telling us to do as many exercises as possible. But when it comes to practical situation, I realize it ...
10
votes
7answers
928 views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...