Linear algebra Definition and 999 Threads

Homework Statement a positive definite matrix has eigenvalues λ=1 and λ=2. find the matrix Homework EquationsThe Attempt at a Solution I've used a 2x2 matrix with entries a0,a1,a2,a3 as the unknown matrix but no use. (As little as i know a0 and a3 should be 1 and 2 respectively...

Homework Statement If n is a positive integer, then 2x2 matrix [-32,252] [-4,32] raised to the power of n is... Homework Equations I know that first I should diagonalize the given matrix, something I also seem to have a hard time with. The Attempt at a Solution I determined the eigenvalues...

Homework Statement Find the matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis Homework Equations I know that for A rotated counter clockwise you use the 2x2 matirx [cos(theta), -sin(theta)] [sin(theta, cos(theta)] and...

Current schedule: Summer: Elective Fall: -not important, but it can't be changed in any way- Spring: Differential Eqs., Classical Mechanics & Mathematical Methods 1 (Physics), Linear Algebra, Chemistry Lab, Elective I'm thinking that since the spring schedule looks kinda heavy I can switch up...

Hello, friends! Let us define the external measure of the set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of ##A## by finite or countable families of ##n##-paralleliped ##P_k=\prod_{i=1}^n...

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are the same? Why? If we have a vector space Q that the dimension is n, can we say that any set of...

Homework Statement Under what restrictions on ##c, d, e##, will the combinations ##c\vec u + d\vec v + e\vec w## fill in the dashed triangle? Homework Equations The Attempt at a Solution Clearly, ##\vec w + a (\vec v - \vec w) + b(\vec u - \vec v)## will be in the triangle when ##0 \leq b...

I was wondering if there was anyone comfortable enough about basic to advanced linear algebra concepts who would be willing to participate in a chat/whiteboard based platform to discuss or answer questions concerning such concepts. I need clarification on a few things

Homework Statement Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is...

Homework Statement Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ). Homework EquationsThe Attempt at a...

Computer science major here. I recently completed calculus I, II, III. Now I know that I'll probably be required to take linear algebra. So far, I expect to do some Gaussian elimination (assuming I know what a row echelon is). What else should I expect from linear algebra?

I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##. In general, such a matrix can be...

Homework Statement Let X=ℝ3 and let V={(a,b,c) such that a2+b2=c2}. Is V a subspace of X? If so, what dimensions? Homework Equations A vector space V exists over a field F if V is an abelian group under addition, and if for each a ∈ F and v ∈ V, there is an element av ∈ V such that all of...

(Not an assigned problem...) 1. Homework Statement pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product It follows that: 2. Attempt at a solution. I think that both of these solutions are...

I'm having trouble getting started on this one and I'd really appreciate some hints. This question comes from Macdonald's Linear and Geometric Algebra book that I'm using for self study, problem 2.2.4. Homework Statement Let U1 and U2 be subspaces of a vector space V. Let U be the set of all...

Homework Statement Find the SVD of Homework EquationsThe Attempt at a Solution I'm stuck My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5...

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https://www.math.ucdavis.edu/~anne/linear_algebra/

I know the arrow -> means a map. For example, defines a linear map. But I cannot figure out what does a X mean? I know that above denote a inner product map. whatever is on the left of the : is defined by whatever is on the right. but what is the x symbol? What is the correct way to read...

Homework Statement Let A be an n×n matrix which has the property that A^2 =A. (i) Write down the most general polynomial in AHomework EquationsThe Attempt at a Solution My biggest problem is that I don't even understand what the question is asking Is it just sum (alphaA^n)=0 but A^n=A...

Homework Statement Show that the following is NOT a vector space: {(a, 1) | a, b, c, ∈ ℝ} {(b, c) Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it. 2. The attempt at a solution I am self-studying linear algebra, and have had a difficulty...

Admit that V is a linear space about \mathbb{R} and that U and W are subspaces of V. Suppose that S: U \rightarrow Y and T: W \rightarrow Y are two linear transformations that satisfy the property: (\forall x \in U \cap W) S(x)=T(x) Define a linear transformation F: U+W \rightarrow Y that...

Homework Statement Let(v1, v2, v3) be three linearly independent vectors in a vector space V. Is the set {v1-v2, v2-v3, v3-v1} linearly dependent or independent? Homework Equations Linearly independent is when c1v1+c2v2+...+ckvk=0 and c1=c2=...ck=0 The Attempt at a Solution c1(v1-v2)+...

Homework Statement I'm told to find the matrix Q of the matrix A Homework EquationsThe Attempt at a Solution So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3 S = [X_1, X_2, X_3] but...

which has higher failure rate: linear algebra or intro to programming. I understand failure rate might be skewed: math majors, who are gifted and interested in math, are required to take linear algebra but not intro to programming, while CS majors are required to take both. Which class do CS...

Hello everyone, I have a question. I'm not sure if it is trivial. Does anyone have ideas of finding a matrix ##A\in M_n(\mathbb{C})##, where ##A## is normal but not self-adjoint, that is, ##A^*A=AA^*## but ##A\neq A^*?##

Homework Statement The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin. Homework Equations $f(x) = x^2 + y^2 + z^2$ $h(x) = x^2 + y^2 = 1$ $g(x) = x + z = 1$ The Attempt at a Solution $\langle 2x, 2y, 2z \rangle...

Homework Statement Let R2 have the Euclidean inner product and use the Gram-Schmidt process to transform the basis {u1,u2} into an orthonormal basis. u1 = (1,-3) u2 = (2,2) Homework Equations Gram-Schmidt process: \\v_1 = u_1 \\v_2= u_2 - \frac{\left ( \left \langle u_2, v_1\right \rangle...

Homework Statement I'm trying to solve this circuit when I try to simulate and run a dc sweep i get that it can't be solved. When I try to find the answer using linear algebra I get this answer after I throw it into MATLAB with the warning “[w]arning: Matrix is close to singular or...

Homework Statement Virtually all quantum mechanical calculations involving the harmonic oscillator can be done in terms of the creation and destruction operators and by satisfying the commutation relation \left[a,a^{\dagger}\right] = 1 (A) Compute the similarity transformation...

Homework Statement I feel like I should know the answer to this, so I believe this to be an easy question. Say have matrix A, and I store the elimination matrices E_1,1 E_2,1 etc. and somewhere in the elimination process I have to use a permutation matrix to swap rows. My question is when I...

Suppose ##x_1(t)## and ##x_2(t)## are two linearly independent solutions of the equations: ##x'_1(t) = 3x_1(t) + 2x_2(t)## and ##x'_2(t) = x_1(t) + 2x_2(t)## where ##x'_1(t)\text{ and }x'_2(t)## denote the first derivative of functions ##x_1(t)## and ##x_2(t)## respectively with respect to...

Homework Statement So I think I found an error in the solution were it attempts to find q_2^ I'm asked find the orthornomal basis for the column space of matrix A. Homework EquationsThe Attempt at a Solution [/B] My question is in what it puts for q_2^ A_2 = [4/3 4/3 -2/3]^T ||A_2|| =...

My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...

Homework Statement I feel like this is a easy question but it seems the answer key doesn't seem to be right. So say I have 2 vectors and I'm trying to find the projection of vector u perpendicular to the vector v Homework EquationsThe Attempt at a Solution So I don't remember doing...

Homework Statement I ask for help in solving the exercises in this project on applied linear algebra. The problem outlined in the project is one in which we are tasked with modeling the demise of a gambler. I need help solving exercise 1 (in red) on page 6. I have pasted the exercise text...

Homework Statement So yeah I'm doing a project were I get to create a problem. I would like to learn how to solve a AC RLC circuit using linear algebra. I'm trying to find all of the currents on the edges of the graphs and find all of the voltages at the nodes connecting the edges. I don't...

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I'll try to be concise. I've been out of math for years and never truly learned to understand it. Until now. I want to put the growth mindset theory to the test and see if I can handle physics (or any STEM field) on a university difficulty. To verify if I'm up to it and even have the slightest...

I am self teaching the subject but I am unsure of what is the whole point and picture

Homework Statement Let A be an n x n matrix, and let v, w ∈ ℂn. Prove that Av ⋅ w = v ⋅ A†w Homework Equations † = conjugate transpose ⋅ = dot product * = conjugate T = transpose (AB)-1 = B-1A-1 (AB)-1 = BTAT (AB)* = A*B* A† = (AT)* Definitions of Unitary and Hermitian Matrices Complex Mod...

So I was trying to learn how to find the angle between two complex 4-dimentional vectors. I came across this paper, http://arxiv.org/pdf/math/9904077.pdf which I found to be a little confusing and as a result not overly helpful. I was wondering if anyone could help at all? Many thanks in...

Homework Statement For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A. The matrices are: ## a) \begin{pmatrix} 1 & 1\\ -1 & 3...

Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...

Homework Statement Given an nxn matrix, if a b exists so Ax=b has no solutions, can A be one-to-one? Homework Equations I understand that as a linear transformation, you need things such as (to be one-to-one as a linear trans) 1. n pivots 2. Only the trivial solution exists to Ax=0 Ax=b...

Hey, I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.

Hello all! I've just started to study general relativity and I'm a bit confused about dual basis vectors. If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...

Homework Statement Derive Cholesky Decomposition for a 3x3 matrix Homework Equations IN: S is Real matrix with dimensions 3x3 and is Symmetric and semi-definite Out: L is a Real matrix with dimensions 3x3 such that S=L*L^t L is lower-triangular The Attempt at a Solution We learned this in...

Homework Statement Does the line through the point P(1, 2, 3) with direction vector d = (1, 2, -3) lie in the plane 2x+y-z=3? Homework EquationsThe Attempt at a Solution From the 2x+y-z i can get the vector (2, 1, -3) and the direction vector, their dot product does not equal zero. So, no it...

Homework Statement Solve the linear system of equations: ax+by+z=1 x+aby+z=b x+by+az=1 for a,b\in\mathbb R and plot equations and solutions in cases where the system is consistent. Homework Equations -Cramer's rule -Kronecker-Capelli's theorem The Attempt at a Solution Using Cramer's rule, we...