Linear algebra is the branch of mathematics concerning linear equations such as:
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{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:
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{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
I'm trying to review some high school maths and work my way to Calculus and Linear Algebra, and I found these three translations of Japanese maths textbooks translated by the AMS and edited by Kunihiko Kodaira. The AMS links to them are:
https://bookstore.ams.org/cdn-1669378252560/mawrld-8/...
So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt.
$$
0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)?
I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
I don't understand the solution: that for (1, ..., 1) the additive inverse is (-1, ..., -1), so the condition is not satisfied (and it is not a subspace).
Which condition is not met?
Thank you.
Hello, PF!
It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background:
- This plan is in preparation for my master’s thesis, I...
So this expression is apparently in Sz basis? How can you see that?
How would it look in Sy basis for example?
The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here?
Thanks
Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$
where $v_1,w_1...
Hello,
I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra.
Some textbooks in my country are quite bad and only teach...
Let's assume that ##A## is unitary and diagonalisable, so, we have
## \Lambda = C^{-1} A C ##
Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##.
I tried using some, petty, algebra to prove that ##C C* = I## but...
Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus?
Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
Actual statement:
Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##.
Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
I’m really unable to solve those questions which ask to find a nonsingular ##C## such that
$$
C^{-1} A C$$
is a digonal matrix. Some people solve it by finding the eigenvalues and then using it to form a diagonal matrix and setting it equal to $$C^{-1} A C$$. Can you please tell me from scratch...
i-th column of ##cof~A## =
$$
\begin{bmatrix}
(-1)^{I+1} det~A_{1i} \\
(-1)^{I+2} det ~A_{2i}\\
\vdots \\
(-1)^{I+n} det ~A_{ni}\\
\end{bmatrix}$$
Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]##
The I-th...
Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##?
Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...
Hi there,
I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove :
Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E##
Then...
Let me first list the four axioms that a determinant function follows:
1. ## d (A_1, \cdots, t_kA_k, \cdots, A_n)=t_kd(A_1, \cdots A_k, \cdots, A_n)## for any ##A_k## and ##t_k##
2. ##d(A_1, \cdots A_k + C , \cdots A_n)= d(A_1, \cdots A_k, \cdots A_n) + d(A_1, \cdots C, \cdots A_n)## for any...
When a matrix is represented as a box it seems all very clear, but this representation
$$
A = (a_{ij} )_{i, j =1}^{m,n}$$
Isn't very representative to me. The i -j thing creates a lot of confusion, when we write ##a_{ij}## do we mean the element of i th row and jth column or the other way...
Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks!
My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, :
Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?)
OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...
(We are working in a real Euclidean space) So, we have to show two things: (1)the arrow goes from left to right, (2) the arrow comes from right to left.
(1) if we're given ##\langle x, y \rangle = 0 ##
$$
|| x+ cy||^2 = \langle x,x \rangle + 2c\langle x,y\rangle +c^2 \langle y,y \rangle $$
$$...
The Homework Statement reads the question.
We have
$$
\langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right)
$$
If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows
$$...
If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##.
Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as
$$
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...
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...
I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
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...
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:
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...
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...
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).
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...