Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Find bases of matrix without multiplying

This question is related to a solved problem in Gilbert Strang's 'Introduction to Linear Algebra'(Chapter 3,Question 3.6A, Page 190). Q) Find bases and dimensions for all four fundamental ...
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Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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84 views

Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...
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49 views

How to construct a matrix for vector-mapping

I have a (probably trivial) mathematical issue which I just can’t wrap my head around. Disclaimer: I’m an engineer, not a mathematician so please excuse any mistakes along the road and my Matlab ...
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3answers
167 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
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314 views

Solve Coupled System of Equations via Matrix

I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows: $-sx+sy=0$ ...
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1answer
89 views

Is the smallest singular value able to measure the similarity between two matrices?

I came across an interesting statement. Given two matrices $A$ and $B$, with orthogonal unit column vectors of the same length. $A$ and $B$ are not necessarily square matrices. One would use ...
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392 views

Maximizing the trace

Say i have the following maximization. $ max_R$ trace $(RZ): R^TR = I_n$ where $R$ is an $n$ x $n$ orthogonal transformational vector. Also, the SVD of $Z = USV^T$. I'm trying to find the optimal ...
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88 views

Find a basis for the subspace of $\Bbb{R}^3$ that is spanned by the vectors

Find a basis for the subspace of $\Bbb{R}^3$ that is spanned by the vectors: $$v_1=(1,0,0), \space v_2=(1,0,1), \space v_3=(2,0,1), \space v_4=(0,0,-1)$$ I am not sure how to solve this problem. I ...
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185 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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10k views

Finding the dimension of subspace span(S)

Problem: Consider the set of vectors $S= \{a_1,a_2,a_3,a_4\}$ where $a_1= (6,4,1,-1,2)$ $a_2 = (1,0,2,3,-4)$ $a_3= (1,4,-9,-16,22)$ $a_4= (7,1,0,-1,3)$ Find the dimension of the subspace $span(S)$? ...
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372 views

Is there any distinction between these products: scalar, dot, inner?

I hope you will forgive a math question that comes up in physics contexts where language is loose. This question migrated from Physics SE. I'm finding that I sometimes don't know what kind of product ...
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453 views

3x3 matrices completely determined by their characteristic and minimal polynomials

How do you show that two 3x3 matrices with the same characteristic and minimal polynomials both conjugate to the same Jordan normal form, assuming no knowledge of the eigenspaces? I know that it is ...
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48 views

Show $V \times \{0\}$ is a subspace of $V \times W$ [closed]

Let $V, W$ be vector spaces. Show $V \times \{0\}$ is a subspace of $V \times W$. I am not sure where to start. I know that $\{0\}$ is the trivial set, but I am not sure how that would relate to $V ...
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75 views

solutions to linear equations involving prime numbers?

Suppose we have the two equations: $2Z - p = Xq$ $2Z - q = Yp$ where $X,Y,Z \in \mathbb{N} $ and $p,q \in \mathbb{P} - \left\{2\right\} $ Are there any solutions where $Z$ isn't prime?
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38 views

Matrix with respect to Basis

In part iii) It doesn't give a particularly wholesome answer and I don't really see where to start with this
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23 views

Interpretation of the basis and coordinates for this solution space.

I found $x_1=x_3$ and $x_2=0$. So, $x_1=t;x_2=0; x_3=t$ Therefore: $$(x_1,x_2,x_3)=t(1,0,1)$$ So, the dimension is $1$ and the basis is $(1,0,1)$. Now, I am having trouble interpreting this ...
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129 views

How do I most efficiently find the perpendicular distance from a point to the convex hull of a collection of circles?

I have a collection of one or more line segments for which I know the (x,y) coordinates of the endpoints. The segments may or may not be parallel and may or may not intersect. Each segment endpoint ...
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73 views

Determine whether V and W are isomorphic?

This was a homework problem that my TA graded but I don't understand why he marked it incorrect. Here is my problem and work: $V=S_{3}$(symmetric 3x3 matrix ) and $W = s_{3}^{\prime}$ (skew symmetric ...
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51 views

Show skew-symmetric, non-degenerate bilinear form $((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) \rangle := \varphi(b)-\psi(a)$

Let $W$ be a finite dimensional $K$ vector space and $W^*$ its dual space. For $V := W \oplus W^*$ the mapping $$ V \times V \to K,((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) ...
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139 views

Inverse of product of matrices

Let $n>m$ and let $A$ and $B$ be $m\times n$ and $n\times n$ matrices. $B$ is invertible. If $A$ was square and invertable, then obviously $$ \left(ABA^T\right)^{-1} = A^{-T}B^{-1}A^{-1} $$ But, ...
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Which, if any, of the following polynomials are in Range(t)?

Let T: P^2 ----> P^2 be a linear transformation defined by T(p(x)) = xp'(x) (i) 2 (ii) x^2 (iii)1-x I was hoping someone would show me how to find the range of one of them so I know how to do the ...
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1answer
37 views

Dimension of solution space has 3 vectors but 6 components?

I am not understanding how this has dimension $3$, but there are six components in each vector. If $3$ vectors span the space, why are there more than $3$ components in each vector? I thought for a ...
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31 views

The solution for the matrix system $(A-X)X=0$

$A$ and $X$ are all matrices. Also, $X\geq0$ element-wise and $A\ne X$. Is $X=0$ the only solution for this nonlinear system $(A-X)X=0$?
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23 views

Error Correction in Matrices

I have a matrix for which I am supposed to find the solutions to Ax=0, however Linear Algebra was some time ago and I cannot remember how to do this. Any help would be appreciated. $A = ...
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1answer
42 views

Change of vector basis

For the bit underlined in orange I understand how to show this result by maths, but i'm struggling to accept it conceptually. $P$ is a change of basis matrix from $E$ to $F$ , so why does P map ...
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1answer
120 views

Centralizer of $SO(n)$

Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$? For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where ...
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24 views

Changing a matrix entry to make it singular given the corresponding entry from U of its LU decomposition

This question is from Gilbert Strang's Introduction to Linear Algebra, 2.2.30 (page 55) If the last corner entry is $A(5,5) = 11$ and last pivot of $A$ is $U(5,5) = 4$, what different entry ...
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61 views

Finding the dimension of $Alt^2(V)$ and $Sym^2 (V)$, given that $V = \mathbb{C}^2$.

The question is quite clear, I think. I know that if I can count the basis elements, then I am done. Here is the information I was given about these two spaces: $Sym^2(V) = < a \otimes b + b ...
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51 views

Reflection of $\mathbb R^2$ about a line $L$

My book gives the following definition: Let $L$ be a one dimensional subspace of $\mathbb R^2$. We may view $L$ as a line in the plane through the origin. A linear operator $T$ on $\mathbb R^2$ is ...
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67 views

Which of the following functions $f : \mathbb{R^2} → \mathbb{R^2}$ is a linear transformation?

Which of the following functions $f : \mathbb{R^2} → \mathbb{R^2}$ is a linear transformation? So I've cross out b and d since they do not work with the zero vector. But both a and c look like they ...
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1answer
79 views

Why is the cross product contained in orthogonal complement?

Let $(V,\langle,\rangle)$ be the $\mathbb R^3$ with the standard bilinear-form and let $W \subset V$ be a two dimensional spanning set given by $v = (x_1,x_2,x_3)$ and $w = (y_1,y_2,y_3)$ and the ...
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34 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
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50 views

Is there a traditional name for the “eigenspace” function?

Let $A$ denote a field, $X$ denote an $A$-vector spaces, and suppose $\varphi : X \rightarrow X$ is a linear transformation. Is there a traditional name for the corresponding "eigenspace" function? By ...
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1answer
86 views

Matrix exponential proof

I am solving a problem: $$A^3=\alpha^2A\implies \exp(A)=E+\frac{\mathrm{sinh}\alpha}{\alpha}A+\frac{\mathrm{cosh}\alpha-1}{\alpha^2}A^2;\, \alpha\in\mathbb{C},\,A\in\mathbb{M}_{n\times ...
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41 views

Uinviersal property of basis of a vector space

Let V be a vector space over a field k. Let B be a subset of V. If any set map from B to any vector space W can be extended uniquely to a k-linear map from V to W. Then B is a basis of V. Can ...
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How to find an equation of the plane, given its normal vector and a point on the plane? [duplicate]

I have a question regarding vectors: Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
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2answers
61 views

Why not $R_j + cR_i \rightarrow R_i$ for Elementary Row Operation of Replacement ? [Lay P6]

P6 of Linear Algebra and Its Applications, 4th Ed by David Lay says: Replacement: $\color{green}{kR_j + R_i \rightarrow R_i}$. For example (from BP P435 Example 2): $\left[\begin{array}{cc|c} ...
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83 views

number of solutions and rank

Consider a matrix A of $a\times b$. If we know the how the rank of the matrix is related to a and b, we can determine (maybe not exactly) the number of solutions for the system. Now, if we know the ...
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How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$

How can I show that $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$ for any natural number $n$ and $x_i \in\mathbb{R}?$ I assume there is something about Cauchy-Schwarz and induction, but I really ...
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how does $\det((\det A) I)= (\det A)^n$

Taking determinants of both sides of $A (\text{adj} A) = (\det A) I$, we have $$\det((A) (\text{adj} A))= \det((\det A) I) \text{ or } (\det A) (\det(\text{adj} A)) = (\det A)^n$$ When looking at the ...
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758 views

Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ ...
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1answer
56 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
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1answer
35 views

Linear Algebra nullSpace and multiplication

If you have a $m \times n$ matrix $A$, and an $n \times p$ matrix $B$ and $AB=0$. Is the $dim (null A) = p$? No clue how to start this.
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Proving That a $1 \times 1$ Matrix has a Rank of $1$

I have an assignment that asks me to prove something but I've hit a roadblock. Let $u$ be a $3 \times 1$ matrix with $u_{1}$, $u_{2}$, and $u_{3}$. Let $v$ be a $3 \times 1$ matrix with $v_{1}, ...
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47 views

Distance of a point to a plane

Let $T$ be the plane $x+2y+3z=11$. Find the shortest distance $d$ from the point $P=(2, 4, 5)$ to $T$, and the point $Q$ in $T$ that is closest to $P$. This is just one of the questions on my ...
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4answers
249 views

If all eigenvalues are 1 or -1, is then $A^{12}=I$?

True or false: If all the eigenvalues of A are either $\lambda=1$ or $\lambda = -1$ then $A^{12}$= I If we have a matrix $$\mathbf A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ this has ...
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1answer
129 views

For all non-zero vectors v in R^n, the non-zero vector u is orthogonal to what?

I have a multiple choice question: ...
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37 views

Books in spectral theory for finite dimensional spaces

I'm looking for beginner books of spectral theory for finite dimensional spaces. I've already heard about this subject, but I don't know where I can find it. What's the domain of this subject? (Linear ...
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1answer
72 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...