Linear algebra Definition and 999 Threads

Homework Statement Calculate the eigenvalues and eigenvectors of the matrix: $$ A= \begin{bmatrix} 3 & 2 & 2 &-4 \\ 2 & 3 & 2 &-1 \\ 1 & 1 & 2 &-1 \\ 2 & 2 & 2 &-1 \end{bmatrix} $$ Homework Equations nothing The Attempt at a Solution I've found the eigenvalues, but what...

Homework Statement if V = C_x for some x belongs to V then show deg(u_L) = dim(V) here, L: V -> V linear operator on finite dimensional vector space C_x = span {x, L(x), L^2(x),....} u_L = minimal polynomial Homework Equations The Attempt at a Solution since...

Homework Statement Let T: C→D with dim(A)=n and dim(B)=m. Show that there exists bases B and B' for C and D, respectively, such that the matrix of T in block form is M=|I 0| |0 0| where I is a k by k identity matrix Homework Equations The Attempt at a Solution Honestly no idea...

Homework Statement find a formula for \begin{bmatrix} 1 & 1& 1\\ 0& 1& 1\\ 0& 0 & 1 \end{bmatrix} ^n and prove it by induction the induction part is ok. I'm just having trouble finding a pattern I may have figured it out but it looks too cumbersome Homework Equations...

Homework Statement A. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity[/color] matrix. B. If the columns of A span R^n, the columns are linearly independent. C. If A is an nxn matrix, then the equation Ax=b has at least one solution for each b...

Homework Statement Let S : U →V and T : V →W be linear maps. Given that dim(U) = 2, dim(V ) = 1, and dim(W) = 2, could T composed of S be an isomorphism? Homework Equations If Dim(v) > dim(W), then T is 1-1 If Dimv < dim(w), then T is not onto. The Attempt at a Solution So...

How difficult will this be? I have 6 semesters left at CC and I need to take college algebra, precalc, calc 1, calc 2, calc 3, linear algebra, and dif equations. So I have to have some overlap somewhere but I am not 100% sure. College Algebra is a prereq for precalc, which I have to take...

I'm doing my homework but I'm lost on one thing. Let's say that we have a systems of equations like so: 2x1+3x2=y1 4x1+2x2=y2 Instead of setting it to a constant our teacher sets it to a variable, he says that to be able to compute this, the augmented matrix should look like: 2 3|1 0 4 1|0 1...

Homework Statement M: V -> V linear operator st M^2 + 1_v = 0 find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v Homework Equations The Attempt at a Solution using M^2 = -1_v, i rewrote the operator(?) as M^3 + M + I_v i don't know what to do. i guessed min poly to...

Homework Statement Show that B|A|+A|B| and A|B|-B|A| are orthogonal. Homework Equations N/A The Attempt at a Solution I'm not too sure exactly how to start this. I do know that for two things to be orthogonal, the dot product has to be equal to 0, but I'm not sure how to evaluate...

Homework Statement If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors Homework Equations llu+vll = llull + llvll cauchy schwarz The Attempt at a Solution (1) llu+vll = llull + llvll (2) (llu+vll)^2 = (llull + llvll)^2 (3) (llu+vll)^2 = llull^2 +...

Homework Statement Here is a picture of the diagram. http://gyazo.com/f1b7051fda5b9e1d3a185c53abde1211 I must use the Gauss Jordan elimination method and solve for X1, X2 and X3 I am having problems setting up my equations Homework Equations The Attempt at a Solution...

Homework Statement Consider the system: (1) 6x + ky = 0 (2) 4x + 6y = 0 The system will have a unique solution when k is: (a) equal to 9 (b) any real number Which statements are true. Homework Equations (1) 6x + ky = 0 (2) 4x + 6y = 0The Attempt at a Solution If m=n (number of equations is...

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the book we're on. Now let me try to reduce some ambiguity in my question, I have a general...

forgive the messiness; i take bad notes in class. http://i.imgur.com/VmW8Ubg.jpg towards the middle of the page where it says "this is equivalent to..." and then my professor wrote what follows but i thought the row vector should be complex conjugates? ie, the red writings are not actually...

so i just wanted to get this confirmed: the only two defined algebraic operations for matrices are addition and multiplication right?

Homework Statement We are supposed to compute the magnitude of vectors that make up a regular hexagon. We are given the magnitude of one side (its magnitude is 1). We are also supposed to compute one of the interior angles. Homework Equations I feel like this isn't enough...

1. I need to prove that for any matrix A(n,n) and a vector v(n,1) the following is true... vTAv=vTATv So far I wasn't able to think of anyway for proving this... any help will be appreciated.

Homework Statement This is a general question in linear algebra Determine whether the following subset of R^3 is a subspace The elements go vertically but I can't show them in this way and will show them horizontally however. ( s-2t, s, t+s ) / s, t ε R ... Homework Equations...

Homework Statement Let V=Pol_3(R) be the vector space of polynomials of degree \leq3 with real entries. Let U be the subspace of all polynomials in V of the form aX^3+(b-a)X^2+bX+(d-b) and W be the subspace of all polynomials in V of the form aX^3+bX^2+cX+d such that a+c-d=0 (i) Does...

so in my book there is an example basically saying that linear transformations can be applied to basis vectors or, more specifically, i think they're using orthonormal basis vectors |e1>,|e2>, ... i'm just a little confused on how they're applying it to the basis vectors. my book...

I am thinking of taking an Intro to Linear Algebra course at a community college to save money while I am at a university. It is a 200 level course just like Calc II and I just was wondering if it was harder than Calc II. I really struggled in that class and I am taking it again this semester so...

Homework Statement Let v_1,...,v_k be vectors in a vector space V. If v_1,...,v_k span V and after removing any of the vectors the remaining k-1 vectors do not span V then v_1,...,v_k is a basis of V? Homework Equations The Attempt at a Solution If v_1,...,v_k span V but...

I know how to solve linear systems but I came across this question where I've never seen the notation before. I searched all over the internet but still couldn't figure it out. The question asked to find all solutions in Z_{2}^{5} of a linear system. I'm guessing that Z^5 means all integers on...

I'd post in the learning materials section but I am unable to do so there for some reason. Which of these is the best to learn from...

Hi, I have 4 implications I am interested in, I think I know the answer to the first 2, but the last two is not something I know, however they are related to the first 2 so I will include all to be sure. Assume that T is a linear transformation from from vectorspace A to B. T: A -> B A* is n...

So as I'm preparing for finals, I'm wondering: The multiplication of two matrices is only defined under special circumstances regarding the dimensions of the matrices. Doesn't that require that compositions of linear transformations are only defined in the same circumstances? I can't...

1. (A)Homework Statement Let the following be A= \left|-1/\sqrt{6} ... 1/\sqrt{3}\right| \left|1/\sqrt{6}... -1/\sqrt{3}\right| \left|2/\sqrt{6}... 1/\sqrt{3}\right| ***excuse the "..." on the matrix, I didn't know how to space them out so I used dots instead*** And the other B=...

The Problem: Let O be the origin and let A, B, C be three points so that the quadrilateral OABC makes an parallelogram. Name (1/4){OA} a, and the diagonal {OB} b. Let P be the point that splits the side OC in the ratio 3 :2 from O. Write the vector {PA} as a linear combination of a and b...

Homework Statement Determine a basis for each eigenspace and whether or not the matrix is defective. \begin{array}{ccc} 3 & -4 & -1 \\ 0 & -1 & -1 \\ 0 & -4 & 2 \end{array} Homework Equations Regular ol' eigenvector, eigenvalue business. The Attempt at a Solution Ok, so I've...

Suppose A is a diagonlizable nxn matrix where 1 and -1 are the only eigenvalues (algebraic multiplicity is not given). Compute A^2. The only thing I could think to do with this question is set A=PD(P^-1) (definition of a diagonalizable matrix) and then A^2=(PD(P^-1))(PD(P^-1))=P(D^2)(P^-1)...

I don't remember exactly how the question on my test was phrased but I believe it was phrased "Let A be an mxn matrix where m>n. Explain why in general there is not a solution to the equation Ax = b where b is a vector in Rm" This question was confusing to me because to me the meaning of...

Hey all, does anyone a great place to learn linear algebra online? Thanks for any help.

Homework Statement x + y+ z = 0 3x + 2y -2z = 0 4x + 3y -z = 0 6x + 5y + z = 0 Homework Equations The Attempt at a Solution I put the equations into a matrix and reduced to RREF. This is what I end up with: x - 4z = 0 y + 5z = 0 The other two rows in the matrix are all...

There, that feels better...

Here is the syllabus for my first linear algebra course: http://gyazo.com/002e551e368990efb32b916dac40c2df Right now, we are going through stuff like operations such as curley E(i,j;lambda) curley D(i,lambda) etc and I don't know where to find extra work on these, so I really need a book...

Homework Statement This is probably a very dumb question, but I just can't wrap my head around what I'm supposed to be doing. The question is: "Determine whether the set is a subspace of R3: All vectors of the form (a,b,c) where a = 2b + 3c" Homework Equations u + v is an...

Hey guys, I'm having problems with a question. Let P be an invertible matrix and assume that A = PMP^{-1}. Where M is M = [{3,1,0}{0,3,0}{0,0,2}] Find a matrix B(t) such that e^{tA} = PB(t)P^{-1}. Now this might be an easy problem, but I really have no idea what to do because my...

Homework Statement let L and M be two symmetric nxn matrices. develop an algorithm to compute C=L+M, taking advantage of symmetry for each matrix. Your algorithm should overite B and C. What is the flop-count? Homework Equations How to minimize the number of flop count? I want to make...

Homework Statement SHow that the set of solutions to a homogenous system of m linear equations in n variabes is a subspace of ℝ^{n} (Show that this set satisfies the definition of a subspace) Homework Equations The Attempt at a Solution If {V1,...Vk}=ℝ^{n} then every vector...

Homework Statement I'm studying for my linear algebra midterm, one of the challenge questions from my textbook is as follows: Using the procedure of Example 8 of Chapter 2.3, find whether or not {(0,1,0,1),(-1,1,4,1),(-1,0,2,2)} is or is not a basis for the hyperplane...

Hi everyone, :) Here's a question I encountered. Don't give me the full answer but explain what is meant by degeneracy in this context. Thank you.

Homework Statement Prove or disprove the following statements. I and 0 denote respectively the identity and zero matrix of the same size as A. If A is a square matrix such that A^2 - 3A +2I = 0 then A-cI is invertible whenever c is not equal to 1 and c is not equal to 2. Homework...

Homework Statement Given the system whose augmented matrix is  1 1 1 1   1 −1 0 a   0 1 b 0  Determine (if possible) conditions on a and b such that this system has (a) no solution (b) many solutions (c) a unique solution. Homework Equations -Row reduction -No solution...

[b]1. The problem statement find the β coordinates ([x]β) and γ coordinates ([x]γ) of the vector x = \begin{pmatrix}-1\\-13\\ 9\\ \end{pmatrix} \in\mathbb R if {β= \begin{pmatrix}-1\\4\\ -2\\ \end{pmatrix},\begin{pmatrix}3\\-1\\ -2\\ \end{pmatrix},\begin{pmatrix}2\\-5\\ 1\\ \end{pmatrix}}...

Hello, How am I to find then number of independent equations in a set using matrix techniques? Thanks

Homework Statement Prove ∥A∥F =√trace(ATA), for all A ∈ R m×n Where T= transpose Homework Equations The Attempt at a Solution I tried and i just can prove it by using numerical method. Is there anyway to prove the equation in a correct way?

Homework Statement Let V = V1 + V2, where V1 and V2 are vector spaces. Define M ={(x1, 0vector2): x1 in V1} and N = {(0vector1, x2) : x2 in V2 0vector 1 is the 0v of V1 and 0vector is the 0v of V2 and 0v is 0 vector of V a) prove hat both M and N are subspace of V b) show that M n N...

Homework Statement 1. Let X be a set and F a Field, and consider the vector space F(X; F) of functions from X to F. For a subset Y\subseteq X, show that the set U = {f \in F(X; F) : f |Y = 0 } is a subspace of F(X; F). NB: the expression \f |Y = 0" means that f(y) = 0 whenever y \in Y...

Homework Statement Let y \inℝ^{3} be a fixed vector, and define T:ℝ^{3}→ℝ^{3} to be Tx = X \times Y, the cross product of x and y. Show that T is linear.Homework Equations The Attempt at a Solution For this question do we have to define another T with the cross product of two other variables...