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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Dual Space Questions

Let $V$ be a finite dimensional vector space over a field $F$. Let $v\in V$ with $v$ not equal to $0$. Show that there is $\varphi \in V^*$ such that $\varphi(v)$ is not equal to $0$. I know ...
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Determine a basis for the solution set of the homogeneous system

Determine a basis for the solution set of the homogeneous system: $$\begin{align*} x_1 +x_2 +x_3 &=0\\ 3x_1+3x_2+x_3 &=0\\ 4x_1+4x_2+2x_3&=0 \end{align*}$$ Then the augmented ...
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2answers
64 views

Calculate explicitly the matrices $M_{E_1}, M_{E_2}, M_{E_3}, M_{E_4}$ such that $[L_A (X)]_\xi=M_A [X]_\xi$

Define for a fixed $A \in \mathbb{M}^{2 \times 2}(\mathbb{R})$ the mapping: $$L_A : \mathbb{M}^{2 \times 2}(\mathbb{R}) \to \mathbb{M}^{2 \times 2}(\mathbb{R}) : X \mapsto AX-XA. $$ Write ...
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366 views

Jordan Canonical Form and Minimal Polynomial

I was wondering what is the relationship between minimal polynomial and the Jordan Canonical Form. Before given a matrix, all you need is to compute the characteristic polynomial to determine the ...
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706 views

Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space

This question is perhaps a little vague; part of what I want to know is what question I should ask. First, recall the following form of the Cauchy-Schwarz inequality: let $V$ be a real vector space, ...
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3answers
99 views

Linear Algebra Question on Eigenvalues

I am having a difficult time with the following question. Any help will be much appreciated. Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
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Another linear algebra question

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,...,a_n$ be the rows of $A.$ Suppose $y=(y_1, ..., y_n)$ is ...
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Matrix Multiplication Interpretation

According to Lecture 3 (from approximately 07:00 - 12:00) of Gilbert Strang's course on linear algebra, Given two matrices $A$, $B$ and their product $C$: 1) Columns of C are combinations of columns ...
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if two vectors are linearly dependent, one of them is a scalar multiple of the other.

Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the other?
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201 views

Different geometrical concepts of vectors

I'm a bit confused about the various geometric concepts of vectors. I'm mainly trying to understand if we can classify any vector into one of two categories.The first category would be free ...
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56 views

Approximate 0 with a integer linear combination

Let $\alpha_i\in\mathbb{R}^m,i\in\{1,...,n\}$ linearly dependent over $\mathbb{R}$. Is it always possible to find Integers $t_1,...t_n, \exists j:t_j\neq0$, such that $\|\sum_{i=0}^{n} ...
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887 views

Rotation matrix in 3-dimensional space with two angles.

I am trying to find a description of a rotation in a three-dimensional space with a matrix that uses only 2 angles. It is easy to find one which uses three angles, since I can always consider the ...
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Lucky Lattice Points [closed]

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
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2answers
197 views

Left inverse of a function

Let $f$ be the function $f\colon \mathbb{N}\rightarrow\mathbb{N}$, defined by rule $f(n)=n^2$. Needed to find two left inverse functions for $f$. I know only one: it's $g(n)=\sqrt{n}$. Does anyone ...
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302 views

Finding all $\alpha$ such that a matrix is positive definite

I have $A = $ $ \left[\begin{array}{rrr} 2 & \alpha & -1 \\ \alpha & 2 & 1 \\ -1 & 1 & 4 \end{array}\right] $ and I want to find all $\alpha$ such that $A$ ...
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3answers
444 views

Special orthogonal matrices have orthogonal square roots

Let $A$ be an orthogonal matrix with $\det (A)=1$. Show that there exists an orthogonal matrix $B$ such that $B^2=A$. Thank you very much.
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4answers
111 views

How do I prove that $v=0$, if $v \in \mathbb{R}^n$ is a vector orthogonal to all vectors $x \in \mathbb{R}^n$?

Suppose that $v \in \mathbb{R}^n$ is a vector orthogonal to all vectors $x \in \mathbb{R}^n$. Prove that $v = 0$.
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How is the permutation matrix found in $A = P^tLU$ factorization?

I am learning how to factorize a matrix into the form $A = P^tLU$, and I am not understanding how the permutation matrix is obtained. A = $ \begin{bmatrix} 0 & 0 & -1 & ...
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2answers
190 views

Linear algebra: need help with proof

Can someone please help me with this proof. For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
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110 views

Matrix inversion help

If $g(n)$ is an integer functions periodic in $a$ And $\phi(a)$ is eulers totient function And $[r_1,r_2,r_3,...r_{\phi(a)}]$ are the postive integers less then $a$ coprime to $a$ With ...
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68 views

Give two linearly independent $X_1,X_2 \in \mathbb{M}^{2 \times 2}(\mathbb{R})$ such that $L_A (X_1) = L_A (X_2) = 0$

Define for a fixed $A \in \mathbb{M}^{2 \times 2}(\mathbb{R})$ the mapping: $$L_A : \mathbb{M}^{2 \times 2}(\mathbb{R}) \to \mathbb{M}^{2 \times 2}(\mathbb{R}) : X \mapsto AX-XA. $$ ...
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1answer
28 views

Value of a linear transformation

$$T(1, -2, 3)=(1,2,3,4)$$ $$T(2,1,-1) = (1,0,-1,0)$$ Find the transformation of $(-8,1,-3)$ Is there a method to use to solve this problem besides just staring at the numbers and trying to figure ...
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1answer
62 views

Finding Determinants Recursively

From the MIT OCW Linear Algebra (18.06) final exam, question 9: For square matrices with 3's on the diagonal, 2s on the diagonal above, and 1s on the diagonal below: $$A_1=\begin{pmatrix} 3 ...
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Solution of a Sylvester equation?

I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
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2answers
117 views

Derivative of $c^TX^TXc$ with respect to $X$

What is the derivative of $c^TX^TXc$ with respect to $X$? Here, all the entries are real and $X$ is a matrix while $c$ is a vector. I keep getting confused with the left and right multiplication. ...
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3answers
93 views

Matrix equation $X + X^T = \operatorname{Tr}(X)A$

Let $A \in M_n(\mathbb{R})$. Solve $X+X^T= (\operatorname{Tr}(X))A $ where the unknown $X$ is in $M_n(\mathbb{R})$. $X^T$ is the transpose of $X$ and $\operatorname{Tr}(X)$ is the trace of $X$.
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143 views

Relation between linear transformations and traces

a) Let $A \in M_n (K)$. We denote $f_A$ the linear form defined, for every $X \in M_n (K)$, by $f_A(X)=Tr(AX)$. Show that the function $f$ which maps $A \in M_n (K)$ to $f_A$ is an isomorphism between ...
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1answer
106 views

In stating that the union of vector subspaces is a subspace iff they are ordered, why require $F$ finite?

On the bottom of page 38 of Roman's Advanced Linear Algebra is written the following (here $V$ is a vector space over the field $F$ and $\mathcal{S}(V)$ is the set of linear subspaces of $V$): "...if ...
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20 views

How to compute effect of change in 1 item's percentage on the others, while maintaining 100%?

Consider three variables x, y, z = 20%, 30%, 50% respectively. Now, let's say i bump up x to 50%. I want to compute the new percentages of y and z to maintain the 100% while in the original ratio of ...
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155 views

Prove that $Au\cdot v = u\cdot A^Tv$

Let A be an $n$ x $n$ matrix and let $u$,$v$ $\in$ $\mathbb{R^n}$. Prove that $$Au\cdot v = u\cdot A^Tv$$ I tried using the fact that $A^Tu=A\cdot u$. However, I cannot seem to get to this result. ...
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6answers
559 views

Let $A^{27}=A^{64}=I$, show that $A=I$

Let $A$ be a square matrix, $A^{27}=A^{64}=I$, show that $A=I$
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2answers
107 views

Find eigenspaces using ruler and compasses

I think this is an interesting question: In the 2-dimensional real vector space, we are given a linear transformation $f$. Suppose we already know the images of the standard bases, say ...
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2k views

Inverse of a partitioned matrix

Suppose I have a partitioned matrix $$\begin{pmatrix} 0 & F^T \\ F & R \\ \end{pmatrix}$$ where $0$ is $k \times k$, $F$ is $n \times k$ and $R$ is $n \times n$. I would much ...
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1answer
97 views

Solve a linear equations with 3 unknowns

I have this equations $$x+3y+6z=3$$ $$x+y+z=-2$$ $$-x+y+4z=7$$ my solution is $$x=0$$ $$y=-5$$ $$z=3$$ this task is 1 of 3 and there is going to be 1 that can be solved, 1 that will have a ...
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1answer
97 views

Property of Division by vector for a field

Serge Lang in "Linear Algebra" on page 2 says that The essential thing about a field is that it is a set of elements which can be added and multiplied, in such a way that additon and ...
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finding the decomposition of Laplacian matrix with position of zero elements unchanged

I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$ $B^TB = A$ where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
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23 views

Show that there are two different ordered bases

everyone forgive the inconveniences I have this problem Let $V$ a complex vector space of finite dimension $T$ a operator over $V$. Show that there are two different ordered bases, $\beta$ and ...
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63 views

Linear algebra mapping question

Does there exist a matrix $A$ such that $$ A\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) = \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right)$$ $$ A\left( ...
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524 views

orthogonal complement of symmetric matrices

How do I can prove that the orthogonal complement of space of symmetric matrices is the space of skew-symmetric matrices? With the inner product $\langle A,B\rangle = \mbox{tr}(A^TB)$. Thanks in ...
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4answers
155 views

How to prove that complex matrices are similar?

How to prove that, for every a in R, the complex matrices are similar? *1*As I understand it sin represents the imaginary number while cos the real. *2*To show similarity there is a need for ...
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1answer
122 views

Eigenvalues and Eigenvectors Diagonilization

Let $ A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$. Ok so the first thing I need to look ...
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1answer
350 views

Geometrical meaning of the Column Space

Suppose I have $2$ planes in $R^3$ and they form a system $Ax=b$. I know the NullSpace of $A$ represents geometrically the vectors that form the intersection between the 2 planes shifted to the ...
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Theoretical problem about basis of a Vector Space [closed]

Let's take the vector space $V$ of $R^2$, that is ... the set of all 2-uples. My first question is : Can we represent any vector contained in $V$ as some linear combination of some basis in $V$? I'm ...
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44 views

Finding an “inverse” of a deviatoric tangent

I have have a material model, defining the deviatoric stress for a nonlinear fluid: $\boldsymbol{\sigma}_{\mathrm{dev}} = f(\dot{\boldsymbol{\varepsilon}}_{\mathrm{dev}})$ Now I wish to find the ...
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4answers
74 views

Are Vectors Orthogonal complement

Let W be the subspace of $R^4$ spanned by (1,0,2,-1) and (3,-2,1,0) is (1,1,1,1) in $W^{\perp}$ is (1,1,-1,-1) in $W^{\perp}$ Please show how to prove these questions
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225 views

Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$

Prove that if $AC^T = |A|I \implies \det C = (\det A)^{n-1}$ Ran into trouble with a proof for linear algebra. $C$ is the cofactor matrix of $A \in \mathbb{R}^{n\times n}$, and I'm not sure how to ...
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400 views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...
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309 views

Finding the distance between the $x$-intercepts of two lines

A line with slope $4$ intersects a line with slope $7$ at the point $(10,28)$. What is the distance between the $x$-intercepts of these two lines? This question was asked in a Math Competition in ...
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1answer
116 views

summing two different orthogonal complements

Let $V$ be an inner product space, not necessarily finite-dimensional, with subspaces $W, X, Y, Z$ such that $W$ and $Y$ are orthogonal complements ($W \perp Y$ and $W + Y = V$) $X$ and $Z$ are ...
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3answers
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Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$ [duplicate]

Let $A$ be a squared matrix, and suppose there exists an $n\in \Bbb N$ in a way that $A^n=0$. Show that $I-A$ is invertible and that $(I-A)^{-1}=I+A+\cdots+A^{n-1}$ I don't have a clue where to ...