Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Show that this map defines a linear transformation.

Show that the map v = [v]E → [v]B for all v ∈ Rn defines a linear transformation TB : Rn → Rn. B = {b1, b2, b3,...,bn} is a basis of Rn. Any vector v ∈ Rn can be uniquely expressed as a linear ...
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Orthogonal vectors to Bivector

If we have set of orthogonal vectors (X) and form that set we create a set of bivectors (Y). Is there a relation among Y based on X? Can we say that Y's are orthogonal as well?
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Prove that the set of points that make up the unit circle are uncountable

My math teacher asked us to prove that the set of points that make up the unit circle are uncountable. We are supposed to do this by "exhibiting" (not sure if this means it can be a proven through ...
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equivalent inner-product vector for one

I have a map that projects a $k$ dimensional vector $x$ is projected to an $m$ dimensional vector $\phi(x)$. I define: $$<x,y>:=\phi(x)^T\phi(y)=\sum_{i=1}^m \phi_i(x)\phi_i(y)$$ Now I'm ...
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Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
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Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
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Let $V=\{f \in X \mid f(0)=f(1)=0\}$ be a linear subspace of $C[0,1]$. Show $(V,\|\cdot\|_\infty)$ is Banach.

Can you please confirm if my proof is correct and if not show where I went wrong. Thanks! Let ${f_n}$ be a Cauchy sequence in $V$ then $f_n(x)$ is a real number for each $x\in [0,1]$ Hence ${f_n(x)}$ ...
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What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
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Proving something is a linear transformation?

If $T:\Bbb R^3 \to \Bbb R^2$ given by $$ T\begin{bmatrix} a\\ c\\ e\\ \end{bmatrix}= \begin{bmatrix} c\\ a\\ \end{bmatrix} $$ then $T$ is a linear transformation: true or false, ...
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The convergence of $f_n(t)$ to $f_n$ in the supremum norm implies $f_n(t)\rightarrow f(t)$ as $n\rightarrow\infty$?

I'm awful at these problems so I was just posting this to confirm whether my solution is correct. As: $$(f_n)_{n\geq 1} \in X \rightarrow f \in X \mbox{ in the supremum norm}$$ $$\mbox{For all } ...
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31 views

General linear group and special linear group

Consider the general linear group $$GL(n,\mathbb R)=\{g\in {\mathbb R}^{n\times n}\mid\det(g)\neq 0\}$$ Prove that the derivative of the function $f=\det:{\mathbb R}^{n\times n}\to\mathbb R$ is ...
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Examples of Unitary Matrices with coefficients all having the same amplitude

I am looking for examples of unitary matrices like this one $$A = \frac{1}{\sqrt{2}}\left( \begin{array}{rr} 1 & 1 \\ 1 &-1 \end{array} \right)$$ where each coefficient has the same amplitude, ...
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Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
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Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
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let A be a $2\times 2$ matrix . Then the smallest number $n\in \mathbb N$ such that $A^n=I$ is

let A be a $2\times 2$ matrix $\begin{pmatrix} \sin \frac \pi {18} & -\sin \frac {4\pi} {9}\\ \sin \frac {4\pi} {9}&\sin \frac \pi {18}\end{pmatrix}$. Then the smallest number $n\in \mathbb N$ ...
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Abstract Linear Transformation Question

I had this question on a quiz today and no idea how to solve it. Please help. Let $ T: \mathbb{R}^n\rightarrow \mathbb{R}^n $ a linear transformation defined by: $\forall \begin{bmatrix}x_1\\x_2\\ ...
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1answer
26 views

Solving a linear optimization problem with products and work benches

I am taking a linear algebra course and I have a homework assignment of: A factory produces 5 products T1, T2, T3, T4, T5. Products are made on 3 different work benches P1, P2, P3, which can be used ...
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Geometrical interpretation of an overidentified linear system

In my econometrics class we talked about Instrumental Variables. Suppose one has a $n\times k$ matrix $X$ of regressors and a $n\times m$ matrix $Z$ of instrumental variables. Given the matrices are ...
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How can we drive frome the matrix charaterization to affine subspace formula?

My question is about the projections, and it consists of two parts: The projection of any point $x\in\mathbb{R}^n$onto any subspace $V$ of a finite dimensional space is defined by ...
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1answer
15 views

Are Primitive Dirichlet Characters linearly independent.

For a positive integer $N$, let $$S_N=\{ \chi~\mid~ \chi \text{ is primitive Dirichlet characters modulo }F,\text{ where } F\mid N \}.$$ I want to check the Linear independence on $S_N$. More ...
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46 views

Surface normal to point on displaced sphere

I want to calculate the surface normal to a point on a deformed sphere. The surface of the sphere is displaced along its (original) normals by a function $f(\vec x)$. In mathematical terms: Let ...
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Max flow min cut from duality

I have been having some trouble deriving the max flow min cut theorem from duality, which I was told is possible. To begin with, I need to cast the problem into the form "maximize $\langle c, ...
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1answer
32 views

How do you find a basis for a polynomial in P2 given a set of polynomials?

I don't know how to show that p1, p2, and p3 actually form a basis for P2. I have been trying different things, but that fixed scalar c has prevented me from forming a basis. .
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Can anyone tell whether this vector space question is true or false? [duplicate]

If U and W are subspaces of a finite dimensional vector space V and V=U+W, then dimV≤dimU+dimW. we know that dimV=dim(U+W) and dim(U+W)> dimU+dimW, and therefore dimV>dimU+dimW. I think this is ...
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1answer
231 views

How to find the exponential of a nilpotent matrix?

I want to find the exponential $e^{tA}$, where $A=\left( \begin{array}{ c c } 0 & 1 & 2 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right).$ I know that its ...
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Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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How do I create a transformation matrix from a polynomial transformation?

I was able to prove it was a linear transformation, but computing the null space has been a challenge. Typically with these problems I would create a transformation matrix based on the linear ...
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2answers
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How can I describe all matrices that can be written as a linear combination of three others?

This is the problem I am having trouble with. I was able to find part a) without much trouble, but part b) has really stumped me. I don't know exactly how to solve for all matrices nor do I know ...
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28 views

How to take apart a characteristic polynomial

Suppose I have a polynomial: $x^3-8x^2+17x-4$. How do I know it will always be $(x-4)(x^2-4x+1)$ by solving it? I'm struggling to figure out what to look for in the polynomial to give me a hint or ...
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29 views

Show/prove that this is a linear transformation

Let A be a matrix of size m by n. Show that T : Rn → Rm given by T(v) = Av for ∀v ∈ Rn is a linear transformation. I don't even know where to begin, for proof problems. I mean, can I just stick ...
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1answer
42 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
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1answer
16 views

Transforming a square matrix A into B

Let's say I have $A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$ and $B= \begin{bmatrix} b_{11} & ...
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1answer
19 views

find the eigenvalue of $A^m$

Let $$A = \pmatrix{7&9\\-3&-5},$$ it is a $2\times 2$ matrix. For every integer $m$, find all eigenvalues of $A^m$, and bases for the corresponding eigenspaces How to get it?!!
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1answer
42 views

If B is invertible, then $\det (B^{-1} AB) = \det A$

The question is if $B$ is invertible then $$\det (B^{-1} AB) = \det A$$ I believe this statement is correct, yet I'm uncertain how to prove it. Would appreciate any suggestions.
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To find dimension of subspace

Let V be subspace of $M_n (R) $ be subspace ofall matrices such that entries in every row add upto zero and entries in every columm also add upto zero .Then i am to find its dimension . I have tried ...
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How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $2-λ [(2-λ)(2-λ) ...
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Second derivative of the position vector in a spherical coordinate system

In a spherical coordinate system my unit vectors are: $\vec{e_r}=\begin{pmatrix}\sin\theta\cdot \cos\phi \\ \sin\theta \cdot \sin\phi \\ \cos\theta \end{pmatrix}$; ...
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1answer
19 views

Eigenvectors of multiplied matrices?

I have the review question if the vector u is an eigenvector of A and and eigenvector of B, then is also an eigenvector of AB, and BA, true or false, and explain why? I just have a feeling its true, ...
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1answer
28 views

Orthogonality of remaining non-intersecting basis

Let $A$ and $B$ $\in \mathbb{C}^{4 \times 100}$ be matrices with null spaces $N(A)$ and $N(B)$ respectively. The dimensions of each null space is $96$ and I was able to find that they intersect in ...
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1answer
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Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that ...
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1answer
28 views

About diagonalization

"Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
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1answer
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Why then is $df(x)$ surjective if and only if $Ax\neq 0$?

$\forall x\in \mathbb R^k$, define the linear map $df(x):\mathbb R^k\to\mathbb R$ as follows:$\forall \xi\in \mathbb R^k, df(x)(\xi)=2x^TA\xi$, where $A$ is symmetric. Why then is $df(x)$ surjective ...
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2answers
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Orthographic projection in euclidean space

Let $E$ be a euclidean space with an inner product given by $$B =\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{array} \right) $$ in a basis ...
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1answer
34 views

Square matrices A and B commute if and only if they share the same generalized eigenspace.

I found a couple of proofs for this theorem but only for the case when A and B are diagonalizable, thus the eigenspace that they share is not the generalized one. Im looking for the proof (or ...
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3answers
27 views

Basis of a vector space is a maximal linearly-independent set?

If $V$ is a vector space of finite dimension over $F$, then a basis of $V$ is a maximal, linearly independent set in $V$. Is this conjecture true? If so, how to prove it?
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Linear Algebra - Weighted Inner Product of Polynomials [on hold]

Given the weighted inner product $\langle f,g\rangle = \int^1_{-1}f(x)g(x)x^2dx$ How do you find an orthogonal basis of the space $\Bbb P^1$ of polynomials of degree $\le$ 1. And how do you find the ...
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2answers
51 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
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Is operator norm invariant under multiplication with orthonormal matrix

Let $A$ be an $r \times r$ real matrix and $Q$ be a $n\times r$ matrix whose columns are orthonormal. I know that $||QA||_2 = ||A||_2$ because $$||QA||_2 = \max_{||x||_2=1} ||QAx||_2\,= ...
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1answer
32 views

Linear Algebra - backward substitution

Is it faster to do back substitution that multiplying A by a vector?
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1answer
21 views

Transformation of coordinate axis to make matrix diagonal

Consider the matrix $$ A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\ \frac{-5}{8\sqrt{3}} & 11/8 \end{bmatrix} $$ which of the following transformations of the coordinate ...