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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
8 views

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 ...
0
votes
0answers
12 views

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: ...
0
votes
0answers
9 views

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 ...
-1
votes
3answers
34 views

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.
1
vote
0answers
15 views

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 ...
0
votes
2answers
25 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\; ...
0
votes
3answers
39 views

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, $x$ ...
0
votes
2answers
16 views

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 ...
0
votes
3answers
62 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 ...
2
votes
0answers
17 views

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 $d=(a_1-a_2$) ...
1
vote
1answer
34 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 ...
0
votes
0answers
14 views

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?
0
votes
1answer
15 views

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 ...
2
votes
1answer
20 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 ...
0
votes
1answer
18 views

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 ...
0
votes
0answers
11 views

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 ...
0
votes
1answer
16 views

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 ...
0
votes
0answers
20 views

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: $A$= ...
5
votes
3answers
116 views

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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
15 views

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 ...
2
votes
0answers
21 views

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 $A,B$ $n\times n$ ...
0
votes
1answer
8 views

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 $u,v\in\mathbb{R}^2$, ...
1
vote
1answer
19 views

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, ...
0
votes
1answer
31 views

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 ...
1
vote
3answers
18 views

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 ...
0
votes
1answer
24 views

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 ...
0
votes
2answers
44 views

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!
0
votes
1answer
20 views

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 ...
0
votes
2answers
26 views

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 ...
0
votes
1answer
14 views

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.
0
votes
1answer
66 views

$e^{(A+B)} = e^Ae^Be^{[A,B]}$ 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 ...
1
vote
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 ...
0
votes
0answers
7 views

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 ...
0
votes
1answer
14 views

Image and Kernel of a Projection of One Line onto Another

The question is: Let T be the projection along a line L1 onto a line L2. Describe the the image and the kernel of T geometrically. I understand that the image should be the Projection of L1 onto L2. ...
0
votes
1answer
34 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...
0
votes
0answers
22 views

Determinant of an elementary matrix

I read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det M ...
0
votes
0answers
8 views

Understanding a proof of RREF uniqueness

Base Case $(n = 1)$: Suppose $A$ has only one column. If $A$ is the all zero matrix, it is row equivalent only to itself and is in reduced row echelon form. Every nonzero matrix with one column has ...
1
vote
1answer
11 views

What is the relation between the cokernel with the kernel of the dual map of a linear transformation?

I am studying linear algebra and I am in front on questions like: What is the relation between the kernel of a linear map and the cokernel of the dual map? What is the relation between theese objects ...
0
votes
1answer
36 views

Define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine ker$\phi$ and im$\phi$

Let {$e_1,e_2,e_3$} be the standard basis for $\mathbb{R}^3$ and define $\phi:\mathbb{R}^3 \rightarrow \mathbb{R}$ by $\phi(e_1) = 1$, $\phi(e_2) = 2$, $\phi(e_3)=-1$. Determine the subspaces ...
2
votes
0answers
21 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
0
votes
1answer
19 views

Show that $Im T$ and $U/Nuc T$ are isomorphic for a linear transformation $T: U \longrightarrow V$

Show that $Im$ $T$ and $U/Nuc$ $T$ are isomorphic for a linear transformation $T: U \longrightarrow V$ Hi guys, I know how to show this for vectorial spaces with finite dimension, but I don't have ...
0
votes
2answers
34 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
0
votes
1answer
20 views

How to construct an isomorphism between $ \ker g^{\ast}$ and $~coker~ g$?

Let $g: L \to M$ a linear transforming. $M, L$ finite dimensional. $g^{\ast} : M^{\ast} \to L^{\ast}$ How do I construct an isomorphism between $ \ker g^{\ast}$ and $coker~ g$? I really don't know ...
0
votes
1answer
36 views

Show that $P_n$ is an $(n+1)$-dimensional subspace of the vector space of all real polynomials [duplicate]

Show that $P_n$ = {polynomials with real coefficients of degree $\leq$ n} is an ($n+1$)-dimensional subspace of the infinite-dimensional vector space of all real polynomials I know that $P_n$ is ...
1
vote
1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...
0
votes
1answer
28 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
0
votes
2answers
36 views

Understanding a problem

Note that these from linear algebra notes. İt was defined fields, showed $\mathbb{Q}$ is a field. Then, below-mentioned qustion was proved. Yet, I didn't ask what happened. Can you explain? What ...
0
votes
1answer
18 views

Number of eigenvalues for this operator

Say I have a F - vector space V and a subspace U given by U={va : a is in F}. Now suppose I have an operator defined by $Tv=av$. Clearly, U is invariant under T, since for any element of U, say bv, I ...
1
vote
2answers
37 views

Unit vectors with imaginary numbers

I'm trying to determine if the matrix: \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix} is a unitary matrix. Therefore, the first step I'm taking is to figure out if both $\langle 0, 1\rangle$ ...