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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Simple explanation for number of solutions of system of linear equations

So a system of linear equations can be represented as: $$Ax=d$$ where $A$ is a $n\times n$ matrix and $x$ and $d$ are $ n\times 1$ vectors. Now in my notes it says the number of solutions are ...
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Proving an orthogonal subspace to v.

Let v be a vector in R^n. Prove that the set is a subspace of R^n (called the orthognal subspace to v.
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342 views

Inverse of a matrix is expressible as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
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257 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
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84 views

What is bi-infinite matrix?

I am reading a wavelet analysis book. In one section there is a term "bi-infinite matrix". I have searched a lot but has not found a good definition. So can anyone tell me, in concise, what's the ...
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A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
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22 views

T-invariance #homework

Let $V$ be a vector space and let $T \in L(V)$. Given a $T$-invariant subspace $U$ is it true that exists a $T$-invariant space $W$ such that $V = U \oplus W $?
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Help finding an eigen vector

Find the eigen vectors of I found $v=\left[1,2/3,1\right]$ and $\left[-1,0,1\right]$ but according to wolfram, there is one more, $\left[0,1,0\right]$, does ...
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showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
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24 views

Help with finding eigenvectors

Find eigen vectors for this: I found that eigenvalues are $0,2,2$ And the eigen vector for $0$ is {$1,0,1$} But I'm not sure how to find the eigenvector for ...
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90 views

Please, help me locate the steps in this induction problem

I am having hard time locating $p(n)$, $p(1)$, $p(n + 1)$, $p(n) \to p(n + 1)$ in the proof below. Please, help me find the borders of induction steps. Thanks. Let $(**)$ $$ \begin{matrix} ...
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Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.
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If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$

How can you show that if $B$ is a maximal linearly independent set of $V$, then this implies that $B$ is a basis of $V$?
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357 views

Generators of Special Linear Groups

Linear algebra and special-linear group experts please help: I learn that in principle one can generate this $M$ matrix form the $B_1$ and $B_2$ matrix below. Here $$ M=\begin{pmatrix} 0& 1& ...
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32 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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70 views

$A$ positive definite iff $BAB^\intercal$ positive definite

I need to prove the following statement: $A$ is positive definite and $B$ is nonsingular if and only if $BAB^T$ is positive definite. Please let me know how this problem would be solved.
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88 views

Find Eigenvalues of multiplied Matrices when the corresponding Eigenvalues are known

I am trying to find the eigenvalues or in particular the largest eigenvalue of a transformation which consists of two matrices: $A = B C$. Assuming I know the EV of both matrices $B$ and $C$, is ...
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30 views

A couple of questions about a proof of the fact that a linear system has a non-trivial solution

Let $(**)$ $$ \begin{matrix} a_{11}x_1 & + & \ldots & + & a_{1n}x_n & = 0 \\ \vdots & & & & \vdots\\ a_{m1}x_1 & + & ...
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Why $\langle y,x\rangle+\langle x,y\rangle=2\mathrm{Re}\langle x,y\rangle$? And the rules of using absolute value, inner production and norm?

Let V be an inner product space over F, x,y∈V. In the proof of triangle inequality, my textbook uses $$\|x+y\|^2 = \langle x,x \rangle + \langle y,x \rangle + \langle x,y \rangle + \langle y,y \rangle ...
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386 views

Gauss-Jordan: Effect of column pivoting on result matrix

When I implement a Gauss-Jordan algorithm I can either have a 1 column result matrix or a multi-column result matrix (I mean the right hand side of the augmented matrix). The first case would be the ...
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22 views

Linear Representations: Show that no $W^0$ exists.

Given the following linear representation and subrepresentation $W$, show that there exists no $W^0$ such that $\mathbb{R}^2 = W \oplus W^0$. Let $\rho: (\mathbb{Z}, +) \to GL(\mathbb{R}^2)$ be ...
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111 views

Existence of $A^2B - BA^2 = 2A \textrm{ and } AB^2 - B^2A = 2B$. in $\mathcal{M}_n({\mathbb{C}}) $

This question arose in this classical exercise : Is there exist two matrices such that $AB-BA=I_n$ in $\mathcal{M}_n({\mathbb{C}}) $. Wich is impossible (by using trace to prove this) But if ...
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51 views

Jordan similar matrix

I have matrix $B = \begin{bmatrix}1 & 1 & -2 & 0\\2 & 1 & 0 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & -1 & 2 & 1\end{bmatrix}$. I found the characteristic polynomial ...
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Is this a basis for the dual space?

There is an example on Wikipedia that I don't understand and I'd appreciate some help. They define $\mathbb R^\infty$ to be the space of all sequences that are zero except for finitely many indexes. ...
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53 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
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Orthogonal vectors and linear systems

Let us suppose we want to solve, with respect to x, the following equation $\mathbf{a}^\intercal\mathbf{b}\;x = 0$ where $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{n} \setminus \{ 0 \}$. It seems clear ...
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119 views

Why is a matrix $A$ that fulfils $AA^t = I$ invertible?

Given a square matrix $A$ that fulfils $$AA^t = I$$ Justify why must $A$ be invertible. The answer, according to my book, is simply $$AA^t = I$$ $$A^t = A^{-1}$$ I don't ...
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35 views

solve this equation in Z

Solve the equation over $\textbf{Z}$ : $x^3$ - 3$y$ = 2 The only way I solve this problem was using the Fermat Theorem. Is there any chance to solve it without using the theorem? And the proof to ...
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22 views

solve the equation in Z

Solve the equation over $\textbf{Z}$ : 2$x^2$ - 2$xy$ - 5$x$ - $y$ + 19 = 0 I tried to obtain some $(A+B)^2$ terms, but I didn't make it. Thanks for your time!
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57 views

Find a basis of the $k$ vector space $k(x)$

Suppose $x$ is a transcendental over field $k$ and $k(x)$ is the field of fractions of $k[x]$. Can we explicitly express a basis of the $k$ vector space $k(x)$?
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43 views

Locus in Complex plane

Could someone help me out with this one Show that the locus of w as z varies with |z| = 1, where w is given by $$w^2=\frac {1-z}{1+z}$$ is a pair of straight lines.
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178 views

Let P be the set of all polynomials of degree ≤ 3 such that p(t) = t. Is P a subspace of P3?

Let P be the set of all polynomials of degree ≤ 3 such that p(t) = t. Is P a subspace of P3? I'm not really sure how to solve this. I know that I have to prove that: Since p, q ∈ P3, k*p ∈ P3 and ...
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106 views

How does Laplace expansion work?

\begin{bmatrix} 1 & 2 & 0 & 0 & a\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 1 & 1 ...
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34 views

is there any such matrix algebra rule

I am following a supposedly easy proof of least square approximation through SVD, but I got confused by this step in the proof. Is there any such matrix algebra rule that justifies this? Thanks a ...
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29 views

Cyclic Supspaces and span

Let $F$ be a field. I am to either prove or provide a counter example to the following: If $T$ is a linear operator on a finite dimensional vector space $V$, then for any $v\in V$, $T$-cyclic ...
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Direct proof : $A.B=xA+yB+zI_2 \implies B.A=x'A+y'B+z'I_2 $

Let $A,B\in M_2(\mathbb R)$. Assume there exist $(x,y,z)\in \mathbb R^3$ such that $A.B=xA+yB+zI_2$. Show that there exist $(x',y',z') \in \mathbb R^3$ such that $B.A=x'A+y'B+z'I_2$ My ...
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44 views

How many iterations are generally required when using the power iteration method?

Suppose I have an n x n matrix and I want to find the dominant eigenvalue and its associated eigenvector. Given these dimensions, what is the minimum number of iterations of the power iteration ...
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61 views

Counterexample to inner product equivalence for complex vector spaces?

I've proved that, if $V$ is a real vector space with a symmetric bilinear form $\langle \ , \rangle$ such that $\langle\overset{\rightharpoonup} v, \overset{\rightharpoonup} v\rangle >0$ for some ...
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232 views

Determine all values of $k$ for which the following matrices are linearly independent in $M_{22}$

If we express these matrix vectors as an augmented matrix, we get a row of zeros. If take out this row of zeros we are left with a $3x3$ matrix, is this allowed? We can find values for which the ...
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Basis of row space equals the basis of a subspace in $\mathbb{R}^n$?

Question: Find a basis for the subspace of $\mathbb{R}^4$ spanned by the given vectors: $(-1,1,-2,0),(3,3,6,0),(9,0,0,3)$ The solution to this problem is the basis for the row space of these ...
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467 views

2x2 Fibonacci matrix singular value decomposition

$A = \left[\begin{array}[c]{rr}1 & 1\\1 & 0\end{array}\right]$ I am supposed to find all the eigenvalues and vectors for this matrix so that $Av=σu$ and then form a singular value ...
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Is a matrix a subgroup of a group when its the inverse matrix “looks different”?

I have the to prove whether a subset of a group is a subgroup. The following subset is given: $$U = \left\{ \begin{pmatrix} a & b & 0 \\ 0 & 1 & c \\ 0 & 0 & d \end{pmatrix} ...
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General Vector Space: Change of basis

If $P$ is the transition matrix from a basis $B'$ to a basis $B$, and $Q$ is the transition matrix from $B$ to a basis $C$, what is the transition matrix from $B'$ to $C$? What is the transition ...
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Find Subspaces of $\mathbb{R}^2$

Find subspaces $A$ and $B$ of $\mathbb{R}^2$ for which the union of $A$ and $B$ is not a subspace of $\mathbb{R}^2$. I'm not sure how to start this!
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41 views

Confusion about endomorphism in two different bases

Let $B=(1,X,X^2)$ and $B'=(P_1,P_2,P_3)$ be two bases of $\mathbb R_2[X]$ where $P_1=X^2+1$, $P_2=X+1$ and $P_3=2X^2-X$. The transformation matrix $P$ from $B'$ to $B$ is \begin{array}{ccc} 1 & 1 ...
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58 views

Showing a $2 \times 2$ matrix that commutes with two non-commuting matrices is a scalar matrix

Let $A, \; B,$ and $C$ be $2 \times 2$ matrices such that $AB = BA$ and $AC = CA$, but $BC \neq CB$. Show that $A$ is a scalar matrix, that is show that for some $k \in \mathbb{R}$, that $A = kI$. ...
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Matrix Rank in Matlab

I want to find rank of matrix in Matlab.I write gfrank(M) in Matlab but Matlab can not find rank of matrix it say matrix is too large to convert to linear index. What can I do?
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32 views

how to prove this equivalence please?

Assume that $E$ is a vector space over $\mathbb K$ (which is $\mathbb R$ or $\mathbb C$). If $(f_{1},…,f_{k}) \in L(E,\mathbb K)=E^*$ (we consider $k$ linear maps from E to $\mathbb K$). I am ...
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33 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
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67 views

Finding a specific basis for an endomorphism

Let $E$ be $\mathbb C$-vector space of dimension $3$. Let $f$ be non zero endomorphism of $E$ such that $f^2=0$. show that there exists a basis $B=\{b_1,b_2,b_3\} $ of $E$ where the matrix of $f$ is ...