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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Exploring underdetermined linear system with non-negative solution

I haven't had much luck searching for this specific problems. Any pointers would be greatly appreciated. I have an underdetermined system where $ A $ and $ b $ are known. $ x $ is a real vector with ...
2
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1answer
159 views

Solving linear equations with Vandermonde

Given this: $$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
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2answers
57 views

Is the matrix V in the subspace U?

I'm given that $U$ is the subspace of $M(3,2)$ generated by $A=\begin{bmatrix} 0 & 0 \\ 1 & 1 \\ 0 & 0 \end{bmatrix}$, $B=\begin{bmatrix} 0 & 1 \\ 0 & -1 \\ 1 & 0 ...
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3answers
132 views

Inverse of a symmetric tridiagonal filter matrix

How to get the inverse of this matrix: $\left(\begin{array}{ccccccc} ...
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1answer
138 views

Closeness of any matrix to a diagonalizable matrix in terms of norm-2

Assuming $X \in \mathbb{C}^{n \times n}$, how to show that for all $\epsilon > 0$, there exist a diagonalizable matrix $D \in \mathbb{C}^{n \times n}$ such that $\left \| X-D \right \|_2 < ...
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0answers
108 views

Transposition of Composition is Reversed Composition of Transpositions

I'm trying to show that $(UT)^*=T^*U^*$. Here is my effort: Consider the following data: \begin{array}{lcl} T:V\rightarrow W & \leadsto & T^*:W^*\rightarrow V^* \\ U:W\rightarrow Z & ...
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1answer
111 views

Proving a diagonal matrix exists for linear operators with invariant subspaces

I came across this problem one of my practice worksheets and I was stumped as to how I would go about solving this. Let $T : V \rightarrow V$ be a linear operator on a finite dimensional vector ...
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1answer
621 views

Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
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1answer
53 views

If basis $\beta$ is orthonormal, then $\beta^{*}=\beta$

Let $V$ be a finite inner product space. Suppose that $\beta$ is orthonormal basis of $V$. How do I show that $\beta^{*}=\beta$? Where $\beta^{*}$ is dual basis of $\beta$ Dual basis definition: ...
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1answer
529 views

Can infinite-dimensional vector spaces be decomposed into direct sum of its subspaces?

I'm reading Axler "Linear agebra done right" and in Chapter 1 he discusses subspaces and direct sum. My question is, are there subspaces of the infinite-dimensional vector spaces, e.g. a functional ...
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1answer
199 views

Rank of a block-triagonal matrix

Given a matrix $C=\left [ \begin{matrix} A & 0 \\ B & A \end{matrix} \right ]$, where rank(A+B)=rank(B), and rank(B)>rank(A), does rank(C)=rank(A)+rank(B) hold? A,B are Laplacian matrices.
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Column space and row space of a matrix of which $\det(A)=0$

If $\det(A)$ not equal $0$, then $\operatorname{Col} A = \operatorname{Row}A$? $A$ is diagonalizable? Thank you a lot.
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1answer
282 views

Dual basis existence and uniqueness.

In Wikipedia, on Dual Basis they say: "Algebraically, a dual set always exists, and gives an injection from $V$ into $V^*$. However, a dual basis exists if and only if a vector space is finite ...
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1answer
702 views

Writing a polynomial as a linear combination of other polynomials

I'm currently working on writing $3(x)_4 - 12(x)_3 + 4(x)_1 - 17$ as a linear combination of $(x)_4,\ldots,(x)_0$ and am having difficulty understanding where the conversion comes from. I have the ...
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43 views

Solution of system of linearly dependent equations.

So, I have the system of equations $x'(t) = Ax$ where $A$ is first row-(4,-2) and second row - (8,-4). This has two eigenvalues, both are 0. But I tried to solve it this way: $x_1' = 4x_1 -2x_2$ and ...
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0answers
47 views

Prove/disprove: if $d \mid f$ and $d \nmid g$ then we can not know if $d\mid (f+g)$ or $d \nmid (f+g)$

Given three polynomials $f,g,d \in \mathbb F[x]$, we need to prove or disprove the following assumption: if $d \mid f$ and $d \nmid g$ so we can not say for sure if $d \mid (f+g)$ or $d \nmid ...
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2answers
580 views

How to interpret “rank” of a matrix intuitively?

What is the physical interpretation of "rank" of a matrix ? Why is it called "rank" ?
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Vandermonde question

I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem): I have ...
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1answer
45 views

How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
2
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1answer
1k views

Intuitive proof of row rank = column rank? [duplicate]

Is it possible to give an intuitive/elementary proof of the theorem that says that the row rank of a (finite-dimensional) square matrix matrix equals its column rank?
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2answers
43 views

Matrix Algebra Manipulation

How does one show with full calculations: $$S=\frac{1}{n}\sum_{i=1}^n(x_i-\bar x)(x_i-\bar x)^T = \frac{1}{n}\left(\sum_{i=1}^nx_ix_i^T\right) - \bar x\bar x^T$$ where $$\bar ...
2
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1answer
160 views

Determinant of matrix?

How can we calculate the determinant of this $\,pn\times pn\,$ matrix. I have tried at my best level, and still am not able to come up with a solution. The matrix $a_{ij}$ entry is defined as $$ ...
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4answers
220 views

Divide and Conquer matrices to calculate determinant.

Do the determinant of a matrix equal to the determinant of submatrices? $$ det\begin{pmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\ a_{21} & a_{22} & a_{23} & ...
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1answer
74 views

Questions about orthogonal matrices.

Let $a_1, b_1, a_2, b_2$ be vectors in $V$ with dimension $n$. Suppose that the lengths of $a_1, b_1$ are the same and the lengths of $a_2, b_2$ are the same. Suppose that the angle of $a_1, a_2$ is ...
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63 views

Converting orientation and speed to position

I have a body located at $(x,y,z)$ at time $t_0$. I know the body is moving at a constant speed, $s$ (I don't know the direction he's moving only the magnitude of the velocity vector). The body's ...
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1answer
140 views

The matrix of rotation

Melvin Schwartz starts Principles of Electrodynamics by matrix of rotations, on page 4: Wikipedia says something similar but less thoroughly, so I will not discuss it. Basically, page says that ...
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1answer
183 views

Proving $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im} T^{*}$

Let V be a finite inner product space with $T:V\to V$ a linear transformation. How can I prove that, $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im}T^{*}$ ? Edit: My purpose is to ...
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2answers
2k views

Solving systems of linear equations using matrices, 3 equations, 4 variables

I understand how to solve systems of linear equations when they have the same number of variables as equations. But what about when there are only three equations and 4 variables? For example, when i ...
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1answer
52 views

Coefficients in characteristic equations of a matrix.

Let $A$ be an $n$ by $n$ matrix over the field of all complex numbers and $\det(\lambda E - A)$ its characteristic equation. Suppose that $$ \det(\lambda E - A) = \lambda^n + c_1 \lambda^{n-1} + c_2 ...
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28 views

Affine maps problems

How to find out a particular affine map when some points are given, say if it takes (0,0) to (1,1), (1,0) to (3,2) and (0,1) to (2,4)?
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1answer
56 views

Matrix with elements of inner product.

I have trouble proving the following: Suppose that $V$ is inner product space. Let $v_1,...,v_n$ be a basis of $V$, and let $w_1,...w_n$ be some vectors. I need to show that, $$G(w_1,...w_n) = ...
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156 views

Self adjoint operator

I am looking in the space of test functions $ \{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\} $whether the n-th derivative is a self adjoint operator. the dot product is given by ...
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2answers
83 views

Vector Spaces expressions

When a vector space is just a set of vectors just like any other linear space . Then why is it that you always need a basis to express vectors ? For example , you don't generally need a basis to ...
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2answers
89 views

Commutators, Unitary Operators

Let $[a, a^\dagger]=aa^\dagger-a^\dagger a = 1$ and $[b, b^\dagger]=bb^\dagger-b^\dagger b = 1$ Show that: $e^{\theta (a^\dagger b - b^\dagger a)}$ is unitary, where $\theta$ is a constant.
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1answer
65 views

a problem on solving a determinant equation [duplicate]

Let $a$ be a real number. What is the number of distinct real roots of the following $$\left| \begin{array}{ccc} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ ...
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1answer
138 views

Does the dim[rowspace] ALWAYS equal dim[columnspace]?

My professor was hinting this was going to be on the exam, but wasn't telling us if this is true. I do believe in fact it is true though, because both the rowspace and column space are determined by ...
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2answers
98 views

Jordan form hard problem

I'm trying to find a matrix $P$ such that $J=P^{-1}AP$, where $J$ is the Jordan Form of the matrix: ...
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1answer
122 views

Number of Invariant Subspaces of a Jordan Block

I'm asking this question on behalf of a person I'm supposed to be tutoring who has this problem as part of eir homework. The problem is "How many invariant subspaces are there of a transformation $T$ ...
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0answers
36 views

Subspace decomposition of the rational canonical form

Let $T:V\to V$ be a linear operator on a $k$-vector space $V$. Is there a nice intuitive description of what the subspaces corresponding to the blocks of the rational canonical form of $T$ are? The ...
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1answer
96 views

$(U\circ T)^{*} = T^{*}\circ U^{*}$

Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
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3answers
88 views

Eigenvalues of $A+\alpha uu^T$

Let $A = diag \left (\lambda_1, ..., \lambda_n \right ) \in \mathbb{R}^{n \times n}$, with $\lambda_1 < \lambda_2 < ... < \lambda_n$. Let $u = \left (u_1, ..., u_n \right ) ^T \in ...
0
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1answer
52 views

How do I explain this in a correct way?

So I was told that I have a $5\times 5$ matrix and one eigenvalue is $2$-dimensional and the second eigenvalue is $3$-dimensional and I have to explain if it's diagonalizable or not. I know that it ...
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3answers
156 views

Is there a faster way to diagonalize this matrix?

I'm asked to diagonalize this matrix for homework: $$\left[\begin{matrix}3&0&0&0\\ 0&2&0&0\\ 0&0&2&0\\1&0&0&3 \end{matrix}\right]$$ But since it's ...
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1answer
122 views

What is a parallel vector space and how do I show it is isomorphic to the solution space?

How can I create an isomorphism between the solution space and a parallel vector space. I'm not sure how to define the vector space and the isomorphism. $$ \begin{bmatrix} -2 & 4 \\ ...
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1answer
40 views

A basis for matrix $A$ is $B$, theorem application?

If a basis for matrix $A$ is $B = \{b_1, b_2, b_3, \ldots, b_n\}$ and $A$ has $n$ distinct eigenvalues $(λ_1, λ_2, λ_3, \ldots, λ_n)$, then $A$ is diagonalizable with $A = PDP^{-1}$, where $P$ has ...
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2answers
47 views

If $null(A)=\{0\}$, how does it relate to eigenvectors?

if $A$ is $n \times n$ and $null(A)=\{0\}$, does it have $n$ linearly independent eigenvectors?
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2answers
86 views

If $X$ and $X-Y^TXY$ are symmetric positive definite, then $\max{|\lambda_i(Y)|<1}$

Consider a symmetric positive definite matrix $X \in \mathbb{R}^{n \times n}$. If there is a $Y \in \mathbb{R} ^{n\times n}$ such that $X-Y^TXY$ is symmetric positive definite as well, how can be ...
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2answers
102 views

Transition formula for 1-forms

$\bf{17.3.}$ Transition formula for $1$-forms Suppose that $(U,x^1,\ldots,x^n)$ and $(V,y^1,\ldots,y^n)$ are two charts on $M$ with nonempty overlap $U\cap V$. Then a $C^\infty \;1$-form $\omega$ ...
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1answer
80 views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ...
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59 views

Does a constant eigenvalue of a linearly parameter-dependent matrix have a constant eigenvector?

Let $A(\alpha)=A_0+\alpha A_1$, with $A_0,A_1\in\mathbb R^{n\times n}$ and $\alpha\in\mathbb R$, such that there exists a $\lambda\in\mathbb C$ with the property that for all $\alpha$: $\det(\lambda ...