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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Linear transformations and range

Let $A$ be an $m \times n$ matrix. Suppose that the matrix equation $AX = Y$ is consistent for any $Y$ that is an element of $R^m$ . (a) What is the range of $T_A$? Justify your answer. (b) Under the ...
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76 views

Leslie matrix stationary distribution

Given a particular normalized Perron vector representing a discrete probability distribution, is it possible to derive some constraints or particular Leslie matrices having the given as their Perron ...
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92 views

Simplify the expression

The below expression has three summations (sigmas) and $L$ is a real-matrix and symmetric, $X$ is a real matrix with $n$ rows and $X_{p\mathbb{.}},X_{q\mathbb{.}}$ denote the $p$ and $q$ rows of ...
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77 views

Decomposition of matrices

Let $A$ be a complex $n\times n$ matrix. Can we always find two vectors $a,b\in \mathbb C^n$ such that $A=a\otimes b^T$?
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Set of border points

I have a problem with solving the following problem: We are given a set that looks like this: $\{x \in \mathbb{R}^n | b \leq a^Tx \leq c\}$ where $a \in \mathbb{R}^n$, $b, c \in \mathbb{R}$ My task ...
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What would be complexity of computing $3^{n^n}$?

Just curious, what would be the computational complexity of computing $3^{n^n}$? I am not sure what it would be like.
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61 views

Can a transformation in different levels of $\Bbb R$ be both onto and one to one?

If $T:\Bbb R^n \to\Bbb R^m$ (where $n\neq m$) is a linear transformation, can $T$ be both one to one and onto? My first instinct was it can, but after thinking about it, it seems the set will either ...
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693 views

Linear algebra proof - invertible matrices

I need to set up a proof for this problem: Given that $A$ and $B$ are both $n\times n$ matrices. $A$ is invertible, and $AB=BA$. Prove that $A^{-1}B=BA^{-1}$. I'm just unsure how ...
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608 views

Orthogonal Complement Proof

Let $W \subset V$ with $\dim V= n$. Suppose $w_1,\ldots,w_m$ is an orthogonal basis for $W$ and $w_{m+1},\ldots,w_n$ is an orthogonal basis for $W^\perp$. a.) Prove that the combination ...
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432 views

Find a basis of $\ker T$ and $\dim (\mathrm{im}(T))$ of a linear map from polynomials to $\mathbb{R}^2$

$T: P_{2} \rightarrow \mathbb{R}^2: T(a + bx + cx^2) = (a-b,b-c)$ Find basis for $\ker T$ and $\dim(\mathrm{im}(T))$. This is a problem in my textbook, it looks strange with me, because it goes from ...
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117 views

Finding x-coordinate of a triangle

If we assume that s1(x)=s2(x) (the areas of the two triangles are equal) and and we know (x1, y1), (x2, y2), (x3, y3)and also we ...
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Polynomial Orthogonal Complement

Let $V = \mathbb{P^4}$ denote the space of quartic polynomials, with the $L^2$ inner product $$\langle p,q \rangle = \int^1_{-1} p(x)q(x)dx.$$ Let $W = \mathbb{P^2}$ be the subspace of quadratic ...
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0answers
101 views

Rank-nullity theorem and binary codes

I am asked to prove the fact that if $C$ is an $[N,k]$ code, and $C^{\perp} = x \in \mathbb{F}_2^N$ $|$ $(x,c) = 0$  $\forall c \in C$, then $\dim C + \dim C^{\perp} = N$. I am regrettably far ...
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74 views

Matrix Norm SVD

Let $||A||_1=tr((A^* A)^{1/2})$ In my linear algebra book, we have the following relations For arbitrary unitary matrices U and V let $||UAV^*||_1=||A||_1$, $||A||_1=\sigma_1+...+\sigma_k$ and ...
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1answer
462 views

matrix of a bilinear form on a space of matrices

For a bilinear form on $ \mathbb R^2$ the matrix of the bilinear form, A, with respect to the standard basis is $a_{ij}= \langle e_i, e_j\rangle$, for $ i =1,2$, j = $1,2$. Then for any two vectors ...
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81 views

Explain Theorem in Linear Map

This is from Axler's Linear Algebra Done Right: 3.20 Proposition If $V$ and $W$ are finite dimensional, then $L(V , W )$ is finite dimensional and dim $L(V,W)=(\dim V)(\dim W)$. Proof: This ...
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291 views

Order 7 matrix with odd entries has determinant a multiple of 64?

if $A$ is $7\times 7$ matrix with all $49$ entries being odd numbers. Show that $|A|$ is a multiple of $64$. You can use the fact that an $n\times n$ matrix with integer entries has an integer ...
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Basic Linear Algebra Proof - Orthogonal Vectors

Prove that if $\mathbf{u}$ and $\mathbf{v}$ are nonzero orthogonal vectors in $\Bbb R^n$ they are linearly Independent. I've struggled with this a bit, here is what I know so far: Suppose ...
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385 views

How to find Basis vectors of a matrix $X$, given basis vectors of its kernel matrix $XX^T$?

If we know basis vectors for $K=XX^T$ (e.g. will be eigenvectors here since $K$ is symmetric), how can we find base vectors for $X$?
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Basic trace inequality

Suppose $A$ and $B$ are self-adjoint matrices. Why is it true that $$Tr(A^2) \le Tr(Ae^{-tB}Ae^{tB})$$ for $t\in\mathbb R$, where $e^x$ denotes the matrix exponential?
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Image of a Linear Operators Eigenspace

Given a Linear Operator T and a vector or linear combination of vectors from within a single eigenspace of T, will those vectors always be projected back into that eigenspace when given as input to T? ...
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172 views

Determinant of the transpose via exterior products

Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant: If $t:U \to U$ is a linear operator and $\dim(U)=n$ then ...
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235 views

Square root of a matrix

For any matrix A, does there exist a matrix decomposition such that: $$ A = Z^T Z $$ A is not necessarily positive definite, so the Cholesky decomposition does not apply. My motivation for asking ...
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884 views

Prove that $A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$ is not invertible

$$A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$$ I don't know how to start. Will be grateful for a clue. Edit: Matrix ranks and Det have not yet been ...
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57 views

Is there a closed form solution for KL between multivariate Gaussian distribution?

I am able to find a closed form solution for univariate Gaussian. However, I am wondering that is there a closed form solution for high-dimensional Gaussian ? Thanks
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159 views

Rotation to principal components under equal variance condition

Suppose a centered configuration $X\in\mathbb{R}^{n\times 2}$ is rotated to two principal components, i.e, $X^TX=diag(\lambda_1, \lambda_2)$, where the diagonal entries are the variance across the $x$ ...
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92 views

Example of algebraic field not honoring the Total Order relation?

I've started studying Calculus. I've stumbled upon the definition of an 'Ordered field'. One of the requirements are for the field to honor the Total Order relation. Meaning that for every a,b, a ...
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2answers
195 views

Not bijective isometry

I want to find a linear isometry $T:V\to V$ such that $T$ is not Bijective. I think that, I need to considere a infinite dimensional space $V$, but I am not sure about these concepts. Thanks for ...
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227 views

For which values of $\alpha \in \mathbb R$ is the following system of linear equations solvable?

The problem I was given: Calculate the value of the following determinant: $\left| \begin{array}{ccc} \alpha & 1 & \alpha^2 & -\alpha\\ 1 & \alpha & 1 & 1\\ 1 & \alpha^2 ...
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442 views

inner product and adjoint operator

This is a problem I found in Schaum's Outlines: Linear Algebra, and I was wondering if someone knew how to solve it. I began using integration by parts, but that approach did not lead to any ...
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Is a projector matrix the inverse of itself?

I want to confirm if a projector matrix is its own inverse. I have $x=Px$ and $Px=P^2x$, so premultiplying the second equation with $P^{-1}$ twice, I get $P^{-1}x=Px$ for all x, implying $P^{-1}=P$. ...
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404 views

converting a parametric R5 vector into a Cartesian form

How do you solve a problem like this. I'm completely stumped. it seems like there should be an easy solution but I'm obviously over looking it. any help would be greatly appreciated.
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1answer
223 views

orthogonal operator preserves Inner product?

Well, Does orthogonal linear map preserves the inner product? I know $T$ is orthogonal iff $||T(x)||^2=||x||^2$, and I know that two vector may not be orthogonal if we change the inner product, I ...
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1answer
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What's the relationship between singular, nontrivial and linear dependent? Basic linear algebra question.

I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 ...
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58 views

How do I show that $S=\{u_1,u_2\}$ is a basis for the solution space $Ax=0$ without doing E.R.Os?

How do I show that $S=\{u_1,u_2\}$ is a basis for the solution space $Ax=0$ without doing Elementary.Row.Operations? Given the solution space of $Ax=0$ has dimension 2. $$A=\begin{pmatrix} 3 & ...
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90 views

How to undo linear combinations of a vector

If $v$ is a row vector and $A$ a matrix, the product $w = v A$ can be seen as a vector containing a number of linear combinations of the columns of vector $v$. For instance, if $$ v = ...
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162 views

Prove : $\rm{col}(B) \subseteq \rm{null}(A)$

Let $A$ be an $m\times n$ matrix and $B$ be a $n\times m$ matrix. Prove that: $$AB = 0 \iff \rm{col}(B) \subseteq \rm{null}(A)$$ Here is the problem in my textbook. I just have a simple solution if ...
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2answers
170 views

Differences between Real Matrices and Complex matrices.

I am going through a course in linear algebra. Most of the time I learn that "this concept can be generalized to complex matrices without loss of generality" or "since it holds for complex matrices, ...
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263 views

Change of Basis and Row Operations

I want to find the shortest way of expressing the following idea so that I can use it as a lemma during exams. I want to execute a change of basis without having to go through the whole process of ...
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74 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
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1answer
40 views

Is $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$ a subspace of $P_{3}$?

Given : $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$ Does: U is a subspaces of $P_{3}$ I think the answer is yes. But in my textbook, they say no. And explain that zero is not in ...
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154 views

Orthogonal Least Squares Proof

Let A be an $m$ x $n$ matrix with $kerA ={0}$. Suppose that we use the Gram-Schmidt algorithm to factor $A = QR$. Prove that the least squares solution to the linear system $Ax = b$ is found by ...
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determinant questions

If $B=M^{-1} AM$, why is $\det B=\det A$? Show also that $\det A^{-1}B=1$. If the points $(x,y,z)$, $(2,1,0)$ and $(1,1,1)$ lie on a plane through the origin, what determinant is zero? Are the ...
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Linear transformation $T$ from $\mathbb{R}^3$ to $\mathbb{R}^3$.

$T$ is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ such that $T(u) = u$, $T(v) = 2v$, $T(w) = 3w$, where $u,v,w$ are non-zero. Then which are these are necessarily true: $det(T) = ...
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How to prove $\det(e^{\lambda_ix_j})\not=0$ where $\lambda_i\not=\lambda_j$ and $x_i\not=x_j$ if $i\not=j$

In try to figure out the exercise: Let $$f(x)=\sum_{k=1}^{n}c_ke^{\lambda_kx}$$where $\lambda_i \not=\lambda_j,i\not=j$,and $c_1^2+c_2^2+\dots+c_n^2\not=0$, then the number of $f(x)$'s roots is ...
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Orthogonal Projection Proof

Let $w_1,...,w_n$ be any basis of the subspace $W \subset \mathbb{R^m}$. Let $A = (w_1,...,w_n)$ be the $m$ x $n$ matrix whose columns are the basis vectors, so that $W = rngA$ and $rankA=n$. ...
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636 views

Solving for four variables with four equations

This seems pretty simple but I am having a hard time figuring it out. I need to find four variables, $\,a,\, b,\, c,\, d,\,$ and I have four equations: 1) $\;u_1a -v_1b + c = u_1' $ 2) $\;u_1b ...
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723 views

To Interpret Solving Systems of Linear Equations Geometrically in Terms of Linear Algebra

I never really understood basic Gaussian elimination & solving systems of equations once I learned some actual linear algebra. I thought this was due to me missing some fundamental aspect of the ...
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700 views

Properties of Self Adjoint Operator (Inner Product)

I can't seem to derive this results that my book "Linear Algebra Done right" is using without explanation. It must be obvious but I don't see it. Let $T$ be a self adjoint operator. How do they go ...
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250 views

Linear combination of two matrices

All matrices I consider are complex hermitian $N\times N$ matrices. None of them are Diagonal matrices. Consider a positive semi definite matrix $A$ and a matrix $B$. I am interested in the sum ...