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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Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
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Understanding basis algorithm result

I've a matrix ${\bf A}$ defined as A = \begin{pmatrix} 1 & -2 & 0 & 3 & 7\\ 2 & 1 & -3 & 1 & 1\\ \end{pmatrix} And ${W_1}$ is the solution ...
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Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
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find when matrix is not diagonalizable

Let $A = \begin{pmatrix} 3&0&0\\ 0&a&a-2\\ 0&-2&0 \end{pmatrix}$ A is not diagonalizable find $a$. how can I tell when $a$ is diagonalizable by it's characteristic ...
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2answers
28 views

It is true that $rank(xy^T)=1$?

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
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Piecewise linear function given three points and two crossover boundaries

Suppose you have three points; $(3500, 700)$, $(52500, 5075)$, and $(527500, 36800)$. As well as two $x$ boundaries $25000$ and $200000$. The question is then to construct three lines (each of which ...
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1answer
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Finite abelian group as Z-module

If $M$ is a finite abelian group then $M$ is naturally a $Z$-module. Can this action be extended to make $M$ into a $Q$-module ?
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Linear trasnformation kernel and image

$V$ is a vector space. Let $T: V \to V$ be a linear transformation. Prove that if $\text{Ker}\: T = \text{Ker}\: T^2$ then $\text{Im}\:T = \text{Im}\:T^2$. How do I prove it?
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Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
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30 views

Image of a linear transformation

Let $T : V \to W$ be a linear transformation. If $A$ is a subspace of $V$, show that its image, $$ T(A) = \left\{ T(x) \in W \mid x \in A \right\}, $$ is a subspace of $W$. I have no idea how ...
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1answer
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To find dimension of $N(A) \cap R(B)$ over R

To find dimension of $N(A) \cap R(B)$ over R A = $\begin{bmatrix} 1 & 2 & 0 \\ -1 & 5 & 2 \end{bmatrix}$ B=$\begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 3 & 1 \end{bmatrix}$ i ...
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Linear independent set of functionals makes certain map surjective

Let $V$ be a finite $n$-dimensional vector space over a field $K$ and $\{\lambda_{1},\ldots, \lambda_{n}\}$ be a linearly independent set of functionals Show that the linear map $$\Lambda:V\to K^n$$ ...
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How Many Times The Clock Hands Make an angle Theta?

You might have encountered such a question where $\theta=90^o$ but this a little different and is causing a little problem. I approached this problem in the following matter and solved for $\theta= ...
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92 views

When $\operatorname{im}(A) = \ker(A)$

Consider the following true/ false qustion: There exists a $2 \times 2$ matrix $A$ such that $\operatorname{im}(A) = \ker(A)$. I know that this is true, but I am not sure how to show it. If $A$ ...
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Proof that $\operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F)$ denotes a group-isomoprhism.

Definition: Let $A$ be a Basis of $V$, $V$ a $K$ - Vectorspace. $M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} $, where $\Phi_A$ denotes the following function: $n := \dim V, \{x_1,\ldots,x_n\} = A$ ...
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1answer
21 views

Anticommuting matrices and their eigenvalues

Let $A,B\in \mathcal{M}_n(\mathbb{C})$. It is known that if $AB=BA$ and $\lambda_1, \lambda_2, \dots, \lambda_n $ are the eigenvalues of $A$ and $\beta_1, \beta_2, \dots, \beta_n$ are the ...
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Unitarily equivalent Triangular matrices

Could anyone help me to prove the following problem? Suppose $(x_1,x_2,\dots,x_n)$ is a permutation of $(y_1,y_2,\dots,y_n)$, then any triangular matrix with diagonal entries $(x_1,x_2,\dots,x_n)$ is ...
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Spliting subspaces and fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
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Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...
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A is a square matrix and given that $A^3 = 2\mathbb{I}$, then show $A-\mathbb{I}$ is invertible and find its inverse [on hold]

Could anyone guide along with this question? I was trying $(A−I)(A−I)^{−1} = I$ and was figuring if there's a way out to expand $(A−I)^{−1}$. I also tried $(A−I)x=0$ but to no avail.
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1answer
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Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
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2answers
28 views

If null(AB) is a subset of null(A), does they have the same rank?

Let $A$ and $B$ be a square matrices. If every solution to $AB_x=0$ is also a solution to $A_x=0$ then $rank(AB)$ = $rank(A)$. I'm not sure if the logic is good here : $AB_x=0 \;\;and\;\;A_x=0\; ...
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Cross product and matrix of rotation

I am looking for simplify the following equation and extract vector $\omega$ to the right side. $(R\cdot x)\times(R\cdot(\omega\times x))$ where $\times$ is the three-dimensional cross product, $x$ ...
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2answers
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Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
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63 views

If $1,-1,0$ are eigen values of $A$ then $\det(I+A^{100})=$?

As the question states, if $1,-1,0$ are eigen values of a matrix $A$ then I need to find what $\det(I+A^{100})$ is. Now I know that $\det A=0$, $\det (I+A)=0$ and $\det(I-A)=0$. But I don't know what ...
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0answers
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dimension and base of arithmetic sequence

Arithmetic sequence is a vector space. But how to find a dimension of it. I try this: arithmetic sequence: $a_1,a_2,...,a_n={a_1,(a_1+d),(a_1+2d),...,(a_n+(n-1)d)}$ With only $a_1$ and $d=(a_1-a_2$) ...
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1answer
35 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
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Find orthogonal projection to x-y, x-z, and z-y, plane

In linear transformation from $R^3$ to $R^3$, how would you find the matrix of the linear transformations to do these projections?
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eigenvalue and rank of a transformation

what i feel is that since the range of the linear transformation is strictly less than $n$ this implies that the transformation is not onto hence the null space contains a non trivial vector.but is ...
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1answer
21 views

Tensor Product of Spaces has Basis of Tensor Products

I am given the following definition of the Tensor Product of spaces Given two vector spaces $V,W$ a vector space S is a tensor product of $V,W$ if there exists a map $M$ $$ M: V \times W ...
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1answer
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Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
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A matrix multiplication problem

Suppose we have been given $2n^2$ vectors $a_1,\dots,a_{n^2}$ and $b_1,\dots,b_{n^2}$ each in $\Bbb Z^{n}$. Form an $n^2\times n^2$ matrix $M$ with $i$th row given by $a_i\otimes b_i$. What ...
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1answer
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Determining if a function is linear, time invariant, both or not

I have the function $y(t)=t^2x(t-1)$ and I need to figure out if it is linear or not and time invariant or not. By the looks of it I guessed it to be not linear but the answer is linear but not time ...
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Writing the space of all possible solutions using a homogenous and particular solution

System of equations: $$\begin{align} 2w + 3x -2y +z &=-1 \\6w+ 10x \quad +6z&=14 \\3w +2.5x -15y -4.5z &= -35.5 \end{align}$$ Particular solution to the system of equations: $A$= ...
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nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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1answer
23 views

Finding a Diagonal Matrix for a Linear Transformation

here is the problem: I am pretty stuck on this one. I thought that the formula for a projection was: wx/ww times w, which in turn forms a matrix [w1^2, w1w2], [w1w1, w2^2] * 1/ (w1^2 + w2^2), but ...
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Intersection of tensor product of vector spaces whose intersection is $\{0\}$ is trivial

Let $V$ and $W$ be subspaces of a finite-dimensional vector space $U$ such that $$V \cap W= \{0 \}.$$ Let $A$ be a second vector space (possibly infinite). Is it true that as subspaces of $A ...
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Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
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1answer
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Additivity Implies Homogeneity of Rational Scalars

I did my best to search for this question on the site- but I did not find it. Here it is: If a function $f:\mathbb{R}^2\to\mathbb{R}$ satisfies $f(u+v)=f(u)+f(v)$ for all $u,v\in\mathbb{R}^2$, ...
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1answer
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If a subspace of a finite-dimensional vector space. Then the subspace is finite dimensional?

I have difficulty in understanding the proof of this statement: Let W be a subspace of a finite-dimensional vector space V. Then W is finite dimensional. The proof goes like this. (Linear algebra, ...
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Area of a parallelogram with three points in $\mathbb{R}^{n}$: $(a,b, 0); (a, 0, b); (0, a, b)$

I have been requested to calculate the area of the parallelogram with three adjacent vertices: $(a,b, 0); (a, 0, b); (0, a, b). First, I have made this diagram: Then I proceed to calculate the two ...
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Condition for right handed invertibility

Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ I'm not really sure how go about this ...
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Find the image of the transformation and write as a span of vectors.

Let $T(a,b)=(a+b,2a-b,3a)$ Find the image of $T$ (as a span of vectors). So I created the augmented matrix and got this: $A$= $\begin{bmatrix}1 & 1 & b_1\\2 & -1 & b_2\\ 3 &0 ...
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2answers
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How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
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What are the $GL_n(F)$-orbits of a group action on the set of idempotent matrices?

Let $S= \{A \in M_{n \times n}(F):A^2=A\}$ (set of idempotent matrices). The general linear group $G=GL_n(F)$ acts on $S$ by $A.g=gAg^{-1}$ (conjugation). I'm having trouble visualizing the ...
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Find $\phi, \psi: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, each nilpotent of order 2, whose composition is idempotent

$\phi: V \rightarrow V$ is nilpotent of order 2 if $\phi \phi$ is the zero endomorphism. Now composition of two such endomorphisms need not be nilpotent of order 2. Find $\phi, \psi: \mathbb{R}^2 ...
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1answer
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Range of sum of vector space

Let $S,T$ be elements of $L(V,W).$ Show that the range$(S +T)$ is a subspace of range$(S)$ + range$(T)$. I tried applying the definition of range, but I wasn't sure how to proceed after that.
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$e^{(A+B)} = e^Ae^Be^{[A,B]}$ for non commuting A and B?

For non commuting A and B, and the derivative of $[A,B] = 0$. Is it true that/how to prove that $e^{(A+B)} = e^Ae^Be^{[A,B]}$ If not, what is the expression according to Wikipedia's article on the ...
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1answer
21 views

Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
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Derivation of variance of a linearly transformed vector

I am trying to derive the variance of a linearly transformed vector. A result was given here. $$ \mathbf{y} = X \, \mathbf{b} $$ $$ \mathbf{b} \sim \mathcal{N}( \mathbf{0}, \sigma^2 I) $$ If we say ...