Linear algebra Definition and 999 Threads

(From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...

given P(-1,1,2), Q(-3,0,4), R(3,2,1), find an equation of the line through P that is parallel to the line through Q and R. All the words after the given three points really confuse me and I just need some help on where to start to tackle this problem. Thanks

Homework Statement Homework Equations N/A The Attempt at a Solution I proved the first part of the question (first quote) and got stuck in the second (second quote). I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...

I am currently brushing on my linear algebra skills when i read this For any Matrix A 1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out ) 2)(A + At)/2 is symmetric Now my question is , why should it be...

Homework Statement Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)## a)Proof that this map is linear b)Find some basis of the kernel b)Find the dimension of the image Homework Equations ##\mathbb{R}_{n}[x]##...

Hello I have a question about the uniqueness of the inverse element in a groupoid. When we were in class our profesor wrote ##\text{Let} (M,*) \,\text{be a monoid then the inverse (if it exists) is unique}##. He then went off to prove that and I understood it, however I got curious and started...

This isn't really a homework question, I just need help understanding the example: ===================== ==================== So transformation takes complex n-tuple as input, and it seems output is also a complex n-tuple (which is what makes it "operator"). But permutations of n entries is...

Homework Statement Prove that each subfield of the field of complex numbers contains every rational number. ' From Hoffman and Kunze's Linear Algebra Chapter 1 Section 2 Homework EquationsThe Attempt at a Solution Suppose there was a subfield of the complex numbers that did not contain every...

Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...

Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...

The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...

Could someone please help me work through the differentiation in a paper (not homework), I am having trouble finding out how they came up with their cost function. The loss function is L=wE, where E=(G-Gest)^2 and G=F'F The derivative of the loss function wrt F is proportional to F'(G-Gest)...

I've been studying graduate level Linear Algebra from Steven Roman's Advanced Linear Algebra (Springer, GTM). It is a terrific book, but many of the concepts are extremely abstract so that I find it difficult to retain what I've learned. Can anyone point me to some books/reading on the...

Hello all Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})## (I already proved that this function is linear) I want to prove that: A regular matrix ##\iff m_A## is an isomorphism. So, here is my approach. Can someone verify whether this is...

I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...

Homework Statement True or False: If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero Homework Equations Definition of Linear Dependence: Vectors are linearly dependent if they are not linearly independent; that is there is an...

I want to understand which of these is computationally expensive (in the sense of computational time) which is more accurate. Also I want to understand which of these two problems (computations time + accuracy) of iterative methods are addressed by multi-grid methods?

Homework Statement Let ##A## and ##B## be square matrices, such that ##AB = \alpha BA##. Investigate, with which value of ##\alpha \in \mathbb{R}## the subspace ##N(B)## is ##A##-invariant. Homework Equations If ##S## is a subspace and ##A \in \mathbb{C}^{n \times n}##, we define multiplying...

Hello PF, I have just finished my first semester in college and did Calc. 3. Now for the spring semester i have to take differential equations and i have been given the advice that linear algebra comes in handy when dealing with DEs. So can anyone recommend a good introduction for linear algebra...

Homework Statement Express the complex number (−3 +4i)3 in the form a + bi Homework Equations z = r(cos(θ) + isin(θ)) The Attempt at a Solution z = -3 + 4i z3 = r3(cos(3θ) + isin(3θ)) r = sqrt ((-3)2 + 42) = 5 θ = arcsin(4/5) = 0.9273 ∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273)) a = -117 b...

Hi! First off, I am actually a math / econ major. I hope I'm still welcome here I am trying to figure out if it's worth it to take both of these courses or just one of them. I have not taken LA before. Course 1: Addition, subtraction and scalar multiplication of vectors, length of vector...

Hello all. I have a question about linear dependency. Suppose we have a set ##S## of functions defined on ##\mathbb{R}##. ##S = \{e^x, x^2\}##. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about. Proof: Let ##\alpha, \beta \in...

Homework Statement The system is a spring with constant 3k hanging from a ceiling with a mass m attached to it, then attached to that mass another spring with constant 2k and another mass m attached to that. So spring -> mass -> spring ->mass. Find the normal modes and characteristic system...

if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1| I couldn't solve this because there is not much given.

The problem I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ## $$ \begin{cases} x = 1 + s - t \\ y = 2 - s \\ z = -1 + 2s \end{cases} $$ The attempt ## s, t ## are the parameters for the two directional vectors which "support" the plane. $$...

I'm a bachelor student in Physics and I would like to continue with a MSc in the field of Condensed Matter Physics. I have to choose between some courses at my university and, since I'm not already an expert in Condensed Matter I would like to have a suggestion. If you were in my situation and...

Hello all. I have a question about a reflexive relation. Consider ##1_V : V \rightarrow V## with ##V## a vector space. Obviously, this is an isomorphism. My book uses this example to show that V is isomorphic with V (reflexive relationship). However, suppose I have a function ##f: V\rightarrow...

Homework Statement Let ## \mathbb{V} = \{0 \}## consist of a single vector ##0## and define ##0 + 0 = 0## and ##c0 = 0## for each scalar in ##\mathbb{F}##. Prove that ##\mathbb{V}## is a vector space. Homework EquationsThe Attempt at a Solution Proving that the first six axioms of a vector...

Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...

Homework Statement Prove that in any vector space V, we have: ##\alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0 \lor \overrightarrow a = \overrightarrow 0## Homework Equations I already proved: ##\alpha \overrightarrow 0 = \overrightarrow 0## ##0 \overrightarrow a =...

Homework Statement We're given some linear transformations, and asked what the null space, column space and row space of the matrix representations tell us Homework EquationsThe Attempt at a Solution I know what information the column space and null space contain, but what does the row space of...

Suppose that I have an overdetermined equation system in matrix form: Ax = b Where x and b are column vectors, and A has the same number of rows as b, and x has less rows than both. The least-squares method could be used here to obtain the best possible approximative solution. Let's call this...

What are the best ones and why for a first-timer like myself (doing self-study)? Thanks very much everyone.

I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?

Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!

So anyone of you know a book that provides a gentle and quick refresher for linear algera, in the spirit of the book "Quick Calculus" by Kleppner and Ramsey? Now that I am studying quantum mechanics, I feel I need to review the linear algebra I studied during my engineering degree. Thanks.

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...

I am taking a linear algebra course and an introductory physics course simultaneously, so I am curious about the connections between the two when it comes to vectors. In beginning linear algebra, you typically study vectors in ## \Re^{2}## and ## \Re^{3}##. Are these the same vector spaces used...

Homework Statement Construct a subset of the x-y plane R2 that is (a) closed under vector addition and subtraction, but not scalar multiplication. Hint: Starting with u and v, add and subtract for (a). Try cu and cv Homework Equations vector addition, subtraction and multiplication The...

This is a linear algebra question which I am confused. 1. Homework Statement Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other". The Attempt at a Solution Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...

Homework Statement The Attempt at a Solution \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ -2 & 1 & -6 & -1 \\ 0 & 2 & 8 & 6 \end{array} \right] \sim \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] From the RREF it is...

the answer key said d is supposed to be 10. but there's a way to evade that row exchange. 1st picture is the question and the 2nd picture is the elimination steps.

So I am constructing an analogy between the self replicating fracturing effect on thin films and the path of a charged particle. (Qualitatively, several cracks have similar shapes to charged particle motion) I won't go into the details of the fracture mechanics, so I will only use E+M...

I am doing a nonlinear least squares estimation on a function of 14 variables (meaning that, to estimate ##y=f(x)##, I minimize ##\Sigma_i(y_i-(\hat x_i))^2## ). I do this using the quasi-Newton algorithm in MATLAB. This also gives the Hessian (matrix of second derivatives) at the minimizing...

(sorry for the horrible butchered thread title... should say "determination", not "determining") 1. Homework Statement In "Principles of Quantum Mechanics", by R. Shankar, a vector space is defined as having dimension n if it can accommodate a maximum of n linearly independent vectors (here is...

Hello, I just completed a first course in linear algebra and really enjoyed my studies. So much so that I want to pursue it more in the fall as an independent study, i am a EE major in college and was curious what directions might be useful for applications in that field.

This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...

Homework Statement Homework Equations a) the one given b) det(A-λI) = 0 find λ values using A c)use λ values to find eigenvectors The Attempt at a Solution This wasn't explained well enough so I can understand it in class. So far, I made the matrix being multiplied to A have the following...

Homework Statement Prove Rθ+φ =Rθ+Rφ Where Rθ is equal to the 2x2 rotational matrix [cos(θ) sin(θ), -sin(θ) cos(θ)] Homework Equations I am having a hard time trying figure our what is being asked. My question is can anyone put this into words? I am having trouble understanding what the phi...

Homework Statement Let P2 be the vector space of all polynomials of a degree at most 2 with real coefficients. Let T: P2→ℝ be the functioned defined by: ##T(p(t)) = p(2) - p(1)## a) Find a non-zero element of the Kernel of T. (I think I figured this one out, but I'm not too sure). b) Find a...