Linear algebra Definition and 999 Threads

Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...

Or is reading a proofs book enough

If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##. Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as $$ x =...

Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...

##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...

Hi this is my first message in this forum , I have this problem in my linear algebra course and I have never seen this type. Let $T : \mathbb{Q}^3 → \mathbb{Q}^3 $ a linear application s.t $(T^7 + 2I)(T^2 + 3T + 2I)^2 = 0$ Find all possible Jordan forms and the relative characteristic...

First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...

Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that: $$ \lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0 $$ where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...

First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps. What we know $$L \ \text{is injective} \iff \ker(L)=\{0\}$$ $$\ker L' = \{ x \in \Im(L) \ | \ L'(x)=0\}$$ $$\Im(L)=\{ x \in V \ | \ \exists \ v \in V \ \text{such that} \ L(v)=x\}$$ Besides, we notice...

Hi, I have a set of ODE's represented in matrix format as shown in the attached file. The matrix A has algebraic multiplicity equal to 3 and geometric multiplicity 2. I am trying to find the generalized eigenvector by algorithm (A-λI)w=v, where w is the generalized eigenvector and v is the...

I have the followinq question: Let ##(,)## be a real-valued inner product on a real vector space ##V##. That is, ##(,)## is a symmetric bilinear map ##(,):V \times V \rightarrow \mathbb{R}## that is non-degenerate Suppose, for all ##v \in V## we have ##(v,v) \geq 0## Now I want to prove that...

We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...

Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...

Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:

Hi guys! :) I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected) I think this one is true, as there is no squared...

We open this lecture with a discussion of how advancements in science and technology come from a consumer demand for better toys. We also give an introduction to Principle Component Analysis (PCA). We talk about how to arrange data, shift it, and the find the principle components of our dataset.

This video builds on the SVD concepts of the previous videos, where I talk about the algorithm from the paper Eigenfaces for Recognition. These tools are used everywhere from law enforcement (such as tracking down the rioters at the Capitol) to unlocking your cell phone.

In this video I give an introduction to the singular value decomposition, one of the key tools to learning from data. The SVD allows us to assemble data into a matrix, and then to find the key or "principle" components of the data, which will allow us to represent the entire data set with only a few

The reference definition and problem statement are shown below with my work shown following right after. I would like to know if I am approaching this correctly, and if not, could guidance be provided? Not very sure. I'm not proficient at formatting equations, so I'm providing snippets, my...

Summary:: Hello all, I am hoping for guidance on these linear algebra problems. For the first one, I'm having issues starting...does the orthogonality principle apply here? For the second one, is the intent to find v such that v(transpose)u = 0? So, could v = [3, 1, 0](transpose) work?

for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection. As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...

The following matrix is given. Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1. The answer sheet reads that the diagonal matrix D is as follows: I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...

What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?

My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution). As the first column is all 0s, x1 would be a free variable. Because the system with free variable have infinite solution, the solution is not unique. In this way, the matrix is...

Hey guys, so I was on this thread on tips for self studding physics as a high schooler with the aim to become a theoretical (quantum) physicist in the future. I myself am a 15 year old who wants to become a theoretical physicist in the future. A lot of people in the thread were saying that...

What are best second course(undergraduate) books in linear algebra for self-study?I have already read Introduction to Linear Algebra by Lang.

So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...

Summary:: Properties of subspaces and verifying examples Hi, My textbook gives some examples relating to subspaces but I am having trouble intuiting them. Could someone please help me understand the five points they are attempting to convey here (see screenshot).

Summary:: What pre-requisites are required in order to learn the textbook "Linear Algebra (2nd Edition) 2nd Edition by Kenneth M Hoffman (Author), Ray Kunze (Author)" Sorry if this is the wrong section to ask what the title and subject state. I read some of chapter 1 already, and that all...

The first thing I do is making the argumented matrix: Then I try to rearrange to make the row echelon form. But maybe that's what confusses me the most. I have tried different ways of doing it, for example changing the order of the equations. I always end up with ##k+number## expression in...

Textbook answer: "If P1P2 = P2P1 then S is contained in T or T is contained in S." My query: If P1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}and P2 =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} as far as I...

%%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...

Im confused as finding the minimum value of lambda is an important part of the proof but it isn't clear to me that the critical point is a minimum

Hi PF community, recently i learned about Calculus in one variables and several, so now i'd like to study linear algebra by myself in a undergraduate level, in order to do that i need some textbooks recommendations. I'll be waiting for your recommendations :).

Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...

0 Let T: P2 (R) → P2 (R) be the linear map defined by T(p(x)) = p''(x) - 5p'(x). Is T invertible ? P2 (R) is the vector space of polynomials of degree 2 or less

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

First time posting in this part of the website, I apologize in advance if my formatting is off. This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!

Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...

I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...

This was a problem that came up in my linear algebra course so I assume the operation L is linear. Or maybe that could be derived from given information. I don't know how though. I don't quite understand how L could be represented by anything except a scalar multiplication if L...

##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?

I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...

I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...

I want to take some courses that involve heavy math, so I have been learning maths on the khan academy site: precalculus, calculus, statistics etc. But one fundamental area of maths the khan academy site doesn't have is a course on linear algebra. I really need to learn and use linear algebra in...

In my book no explanation for this concept is given and i can't find anything about it when I am searching. One example that was given was: Let $$A=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$ with ##k=\mathbb{Z}_2## I think k is the set of scalars for a vector that can be multiplied with...

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...