Linear algebra Definition and 999 Threads

Homework Statement What is the relation between the image of A and the image of A2 + A? Homework EquationsThe Attempt at a Solution im (A^2 + A) for x (A^2+A) is within the image. Linear combination properties show A^2 x + A x. Not sure where to go from here

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...

Homework Statement Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA. Homework EquationsThe Attempt at a Solution I think I have this figured out, just checking it. Heres what I got: ABC=I (ABC)B-1=IB-1 (B*B-1)AC=IB-1 I*AC=IB-1 Cancel I using left...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

Homework Statement Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1 Homework EquationsThe Attempt at a Solution At first I started with the property A3=A I then applied the determinant to both sides. From this point I don't really see any...

Recommend a self study book for linear algebra with complex numbers

How do I prove that this problem doesn't have a solution? 1. Homework Statement 12. Consider the Leontief system Here the column sums are 1, violating the Leontief input constraint given in the text. Show that this system cannot have a solution. Hint: Add the three equations together...

Homework Statement The vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a ____. Homework Equations The Attempt at a Solution This is really straight forward, but I cannot validate the answer to myself. The textbook says that they should lie on a line, but why is this? Obviously if...

I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?

Hi all, I am taking Quantum Mechanics II this coming spring semester. However, I haven't taken Linear algebra yet but I don't want to take the class just because for some few topics. What linear algebra topics should I learn before the class? Thanks!

My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector spaces. It is as follows: Let W=W(n,\mathbb{R}) be a vector space with a basis...

Homework Statement A line is perpendicular to the line 2x - 4y + 7 = 0 and that passes through the point P(7,2). Determine the equation of this line in Cartesian form. Homework Equations n/a 3. The Attempt at a Solution Okay, so how I generally believe how to solve these problems may be...

for this question i got up to obtaining [1 0 1 a-c; 0 1 1 c; 0 0 0 b-a+c] matrix im supposed to find a b and c so that the linear system has an infinitely many solutions I just can't work this out!

I am working through linear algebra from MITs MOOC online courses. One of the question refers to the uncertainty principle. It states: AB-BA=I can happen for infinite matrices with A A=A^{ T }\\ and\\ B=-B^{ T }\\ Then\\ x^{ T }x=x^{ T }ABx-x^{ T }BAx\le 2\parallel Ax\parallel \parallel...

Dear Physics Forum personnel, I am a college sophomore with double majors in mathematics and microbiology. I apologize for this interruption but I wrote this email to seek your advice and recommendation on linear algebra textbook. I will be taking the "theoretical, proof-based" introductory...

Homework Statement For u and v in R^n prove Minkowski's inequality that \|u + v\| \leq \|u\| + \|v\| using the Cauchy-Schwarz inequality theorem: |u \cdot v| \leq \|u\| \|v\|. Homework Equations Dot product: u \cdot v = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n Norm: \|u \| = \sqrt {u \cdot u}...

I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...

I am currently studying A level Further Pure Mathematics, and I will be joining university next year. I want to brush up on precalculus (algebra, trigonometry, geometry) and calculus (differential, integral [single variable]), except that this time I need a more rigorous treatment of those...

I am planning my courses for next semester, and I am unsure of which direction to go in with linear algebra. I have already taken the following course: Linear Algebra I Properties and applications of vectors; matrix algebra; solving systems of linear equations; determinants; vector spaces...

Homework Statement Let Hom(V,W) be the set of linear transformations from V to W. Define addition on Hom(V,W) by (f + g)(v) = f(v) + g(v) and scalar multiplication by (af)(v) = af(v. If V is a vector space over a field K, define V* = Hom(V,K). This is called the dual space of V. If...

Is it possible to obtain a solution of the linear system Ax = b with LU decomposition when A contains zeros on its diagonal? I am trying to obtain a solution with LU decomposition and then perform a forward/backward substitution but I get NaN entries in the solution vector x. The condition...

Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...

Homework Statement Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>. Homework Equations Hey everyone, so I'm a physics student who had to choose a few electives in...

Homework Statement How do I find a basis for: the subspace of R^3 consisting of all vectors x such that x ⋅ (1,2,3) = 0. Homework Equations I believe this is performed through setting x = x,y,z, setting each parameter sequentially equal to 1 while the others are set to o, putting into a matrix...

Homework Statement How to indicate that a vector b is in the span of the columns of a matrix C? Homework Equations I could type the definition of Span here, but Wikipedia has it too and it is not necessary or useful now. The Attempt at a Solution \mathbf{b} \in \mathrm{Span}\{\mathbf{c}_1...

I am in Calculus BC in high school right now and I am really enjoying it and have finished all of the material for the year and I have heard different things about what class comes after Calculus BC (I believe BC is equivalent to Calc 1 & 2 in college). I have heard the next class, aka Calc 3...

Hi all. I was hoping I could clarify my understanding on some basic notions of dual spaces. Suppose I have a vector space V along with a basis \lbrace\mathbf{e}_{i}\rbrace, then there is a unique linear map \tilde{e}^{i}: V\rightarrow \mathbb{F} defined by \tilde{e}^{i}(\mathbf{v})=v^{i}...

Homework Statement As a string in my program. Homework Equations Solving a system with the forward phase of row echelon reduction and a consecutive back substitution. All done by numpy here. (The book suggested MATLAB, etc). The Attempt at a Solution import numpy """ In a wind tunnel...

Homework Statement Find Kernel, Image, Rank and Nullity of the matrix 1 −1 1 1  | 1 2 −1 1 | 0 3 -2 0 Homework EquationsThe Attempt at a Solution I have reduced the matrix into rref of 3 0 1 3 0 3-2 0...

Homework Statement I could prove a, trying b now. Homework Equations The definition of the cross prod.? The Attempt at a Solution https://www.dropbox.com/s/0sauaexkl4j2yko/proof_cross_prod.pdf?dl=0 I did not manage to get a scalar times v and a scalar times w. (No need to point this...

Homework Statement Let \Delta be a line and A a point in space. [itex ] B [/itex] is a point on \Delta ans H is a point on \Delta such that \newcommand{\vect}[1]{\vec{#1}} \vec{AH} \perp \Delta . a) Show that \| \vec{AH} \| \leq \| \vec{AB} \| (a geometric reasoning should be used)...

1- How can infer from the determinant of the matrix if the latter is real or complex? 2- Can we have tensors in an N-dimensional space with indices bigger than N?

Homework Statement In Exercises 45-46, show that the plane and line with the given equations intersect, and then find the acute angle of intersection between them. 45. The plane given by x + y + 2z = 0 and the line given by x = 2 + t y = 1 - 2t z = 3 + t Verbatim from Poole - Linear Algebra: A...

Homework Statement Hello there everybody! I'm reading a Linear Algebra textbook, specifically on LTV systems solutions. I'm trying to redo this example from the book: Homework Equations But I couldn't understand the passage: The Attempt at a Solution I mean. x1(0) = 1 and x2(0) = 0? I...

Hi all, So I'm going to have my first exposure to linear algebra and I've completed Calc 1 and 2. I've seen Axler and Shilov numerous times and I'm having a hard time choosing it. Here's my syllabus for my Linear Algebra Course. Matrices, Gauss reduction, invertibility. Vector spaces, linear...

Consider the linear transformation T: R3 --> R3 /w T(v1,v2,v3)=(0, v1+v2, v2+v3) What is the preimage of w=(0,2,5) ?I tried setting up the system of equations and got v1+v2= 2 and v2+v3=5 but after that I got kinda lost in how to find the individual solutions?

Homework Statement let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis Homework Equations find the matrix A The Attempt at a Solution I got stuck on rotating z component. I tried T([e1,e2,e3])=[0 -1 0]...

Homework Statement Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span S = [ (2,0,3) , (2,0,-1) , (6,0,5) , (4,0,6) ] Homework EquationsThe Attempt at a Solution I know S does not span R3 because the system of...

Okay, long story short, my mom could not work because of an illness so me and and family all had to chip in so fulfill the mortgage payment on our home. I worked two jobs and am a full-time student. I've been studying as much as I possibly can and may have to temporarily restrain from spring...

Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) )Note: The...

How would you prove that adding two vectors in the column space would result in another vector in the column space? I know this is maybe the most basic property of vectors and subspaces, and that the very definition of the column space says it's spanned by vectors in the column space. Is there...

If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of...

Find the matrix for the transformation that projects each point in R3 (3-D) perpendicularly onto the plane 7x + y + 3z = 0 . The attempt at a solution is attached for question 1 (actually instructor's solution) I kind of understand it but ... why is n <dot> v = equation of the plane? Does v...

Here is a picture of the problem. Can anyone give me some hints on the problem? I've looked in my textbook, but I don't know what "s" means. I found stuff on the parametric vector form, and it gives me the equation x = su + tv, but I don't see any "t"'s in this problem. I first tried...

As I study today, I read through my textbook that says if Ax=b has non-trival solution, then b span in A. I want to know how can I prove it.

I have some measurement data for known spectrum. Now I want to establish a algorithm for later measurement of unknown spectrum. I attached the equation in the image. r g b filters apply to the true spectrum I I am looking for. R G B are the measurement getting from the device. I1, I2, I3 are...

Hello all, I've got one more semester before I earn my physics MS, and I have space for one or two extra courses. I am going into oceanography, and I would like to have a strong foundation in math in order to understand the theory I'll encounter as well as possible. Lots of physical...

I've learned that a vector in coordinate system can be expressed as follows: A = axAx+ayAy+azAz. ai, i = x, y, z, are the base vectors. The transformation matrix from cylindrical coordinates to cartesian coordiantes is: Ax cosΦ -sinΦ 0 Ar Ay = sinΦ cosΦ...

the first row 1 0 0 2 the 2nd row 0 1 2 0 the 3rd row 0 2 1 0 the 4th row 2 0 0 1 I would like to ask which is the most efficient way of solving this ques.Though i can solve but is long method, I know there must have some quick 1, appreciate if u can share it. thank you

Homework Statement find the state transition matrix of a time varying system where: dX/dt = A*X with A = [-1 , exp(-t - (t^2)/2) ; ; 0 , t] (Matlab format - sorry but its easier) Homework Equations How to go about solving such problems in a systematic way? The Attempt at a Solution...