Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Linear independence of finite subset

Prove that a set $S$ of vectors is linearly independent if and only if each finite subset of $S$ is linearly independent. My progress: I have been able to prove that if $S$ is linearly independent ...
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Find initial system from solution of a RREF matrix.

I was looking at the solution's of my exam. I used a different technique: I first found the RREF of the matrix giving the solution. Then I worked backward to find the initial system. Can anyone ...
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Consistent Augmented Matrix

Well, the linear system which at least has one solution is called "consistent" linear system. Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system: ...
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3answers
113 views

Is $A$ invertible if $ABAB^2 = I$?

Q: Have two matrices of order $5$, $A$ and $B$. If $ABAB^2 = I$, is $A$ invertible? A: Yes, the inverse of $A$ would be $BAB^2$ My definition of an inverse matrix is: For some matrix $X$, ...
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Closed form solution

I have the following optimization problem: $$\min_{\mathbf{G}} \|\mathbf{B(A+G)\|_F^2} \quad{} \\\text{subject to} \quad{} \mathbf{\|C^T(A+G)\|_F^2\leq \gamma \|A^T(A+G)\|_F^2 } \quad{}, \\ ...
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3answers
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Why does a matrix column being a multiple of another imply that the matrix is singular?

Q: If a matrix of order $9$ has a column that is a multiple of another, does the system $Ax = 0$ has infinite solutions? A: Yes, because the matrix would not be invertible. Can you ...
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49 views

What must I do with this Linear operator?

Can anyone help help me with this problem I want to show that the operator $\mathscr{L}=u \partial/\partial y$, is not an linear operator. This is what i got $\mathscr{L}(au+bv)=u \partial/\partial ...
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2answers
57 views

Linear independence of matrices $I, A, A^2$

I want to prove that $I,A,A^2\:$matrices $\in M_{2\times 2}$ are $\textit {linearly independent}$. I consider the following matrices and their "corresponding" vectors: $I=\begin{pmatrix} 1 & 0 ...
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4answers
38 views

Linear algebra span question?

Let $U$ be the vector $ \begin{bmatrix} 2\\ -1 \end{bmatrix} $ and let $V=\begin{bmatrix} 2\\ 1 \end{bmatrix}. $ Show that the \begin{bmatrix} h\\ k \end{bmatrix} is in the $\text{Span}\{U,V\}$ for ...
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How demonstrate a symmetric tensor in eigenframe?

I am trying demonstrate the following equation (Tsinober Equation 4.1), $$ \begin{equation} -\frac{\partial{Z}}{\partial{x_{i}}}\frac{\partial{Z}}{\partial{x_{j}}}S_{ij}=-\left(\nabla ...
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Find subspaces $U, V, \text{and} \ W$ such that $U \cap (W + V) \neq (U \cap W) + (U \cap V)$

Find subspaces $U, V, \text{and} \ W$ such that $U \cap (W + V) \neq (U \cap W) + (U \cap V)$ This first part of this question was to find subspaces such that the equality holds true. For that I used ...
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1answer
14 views

Diagonalization of matrix using change of variables

In linear algebra, we know that a system of equations $AX=b$ can be easily solved if $A$ is found to be of diagonal nature. If however $A$ is not diagonal but can be changed into a diagonal form by ...
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2answers
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Hey guys. Given the graph below, find the equation of the transformed parent function. [on hold]

It would be great if there is a detailed explanation. Also, is there a standard method I can use to answer all kinds of graphs including exponents and logs? Thanks
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Proof check: any linear transformation can be represented as a matrix-vector product

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf ...
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3answers
74 views

Showing that a set is a basis of a field as a vector space over a subset of that field

Let $K \subseteq L \subseteq F$ be fields and assume that $\{\alpha_1,\ldots,\alpha_m\}$ is a basis of $F$ as a vector space over $L$ and $\{\beta_1,\ldots,\beta_n\}$ is a basis of $L$ as a vector ...
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17 views

Block form of QR factorization

My question is simple. Given a matrix $M$ in the block form: $$M=\left( \begin{array}{cc} A & B \\ C & D \end{array} \right)$$ where $A$ and $D$ are square. Is there an useful block form ...
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18 views

Clarification on Some Definition of Inner Product Space

Suppose $V$ is finite-dimensional Real vector space and $T\in \mathcal{L}(V)$. Suppose that $V$ has a basis $(e_1,e_2,..,e_n)$ of eigenvectors of $T$, every element of $V$ can be written as a linear ...
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29 views

smallest and largest eigenvalue of discretized operator $-d^2/dx^2$

In 1D, the second order derivative operator $-d^2/dx^2$ can be discretized as, using Matlab ...
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11 views

Show that $V = {f ∈ R[t] | f(−a) = −f(a) ∀a ∈ R}$ is a vector space by scalar multiplication and addition [on hold]

How would i show this? would i do there exists and $f, g ∈ R[t]$ and continue? Also the mark scheme added that V consists of polynomials having uniquely terms of an odd degree. How did they find this ...
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11 views

Help with self adjoint operator.

I have a self adjoint operator $h$ acting on vector fields $X$ in $\mathbb{R}^3$. Let us write $X=X_1\partial_{x_1}+X_2\partial_{x_2}+X_3\partial_{x_3}$, where each $X_i$ is a polynomial of certain ...
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35 views

Determinant Question (Proof)

Let $C$ and $D$ be $n \times n$ matrices where n is odd such that $CD = -DC$. Show that either $C$ or $D$ has no inverse. I have no idea how to go about doing this problem. Any help would be ...
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Seam Carving - Energy functions. How do they work?

I have been taking an interesting in dynamic programming and more specifically Seam Carving. For those who are not aware what this is, please look here and for some more detailed information here. If ...
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27 views

Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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1answer
19 views

Checking whether root exists in of a matrix quadratic equations.

Consider a polynomial $X^d+X^{d-1}+...+C=0$ where $X$ is a matrix whose entries are from the finite field $F_p$ and $C$ is also a matrix from $F_p$. How to verify the equation has roots in $F_p$. If ...
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2answers
99 views

Can the transpose of a matrix be expressed in row/column operations?

Suppose that $A$ is a matrix, can we get its transpose, $A^T$, by performing row and/or column operations to $A$?
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1answer
15 views

Transformation Matrix of a linear function

Consider the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$. Let $A = \{ (1,2,3)^t, (1,0,4)^t,(0,0,2)^t \}$ a base of $\mathbb{R}^3$ and $B = \{ (1,1)^t , (2,1)^t) \}$ a base of $\mathbb{R}^2$. ...
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1answer
28 views

Determinant and generalized eigenvalues

Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for ...
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1answer
18 views

Unique linear combination problem

Let $V$ be a vector space and $S$ a subset of $V$ with the property that whenever $v_1,v_2,\ldots v_n\in S$ and $a_1v_1+a_2v_2+\ldots a_nv_n=0$, then $a_1=a_2=\ldots=a_n=0$. Prove that every vector in ...
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Does $\forall x\ne 0: x^TAx>0$ means all eigenvalues of $A$ are real?

Let $A\in\mathbb{R}^{n\times n}$ Does $\forall x\ne 0,x\in \mathbb{R}^n: x^TAx>0$ means $A$ has only real eigenvalues (roots of the characteristic polynomial are all real)?
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2answers
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Eliminating $t$ from the equations, $x=\cos t$,and $y=2 \sin t \cos t$ ,we get [on hold]

Eliminating $t$ from the equations, $x=\cos t$,and $y=2 \sin t \cos t$ ,we get Note: $$\cos^2t+\sin^2t+2\sin t\cos t=0$$ $$x^2+y+\frac{y}{2x}=0$$
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0answers
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Proof for a rank-one Decreasing step

I came across this result in a paper in my area concerning rank-one decompositions. However, I am unable to understand one of the steps. I am reproducing the result here. Let $X$ be a $N\times N$ ...
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Show that $\pm 1$ are the only eigenvalues of the linear operator $f$ as the transpose of a matrix

Let $V$ be the vector space of $n\times n$ matrices over $K$ under addition and let the linear operator $f$ be given by $f(A)=A^{T}$, where $A^T$ denotes the transpose of matrix $A$. Show ...
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1answer
26 views

Positive Semidefinite Matrix

$A$ and $B$ are both positive semidefinite matrices, define matrix $C$, and $C(i,j)$ = $A(i,j) \times B(i,j)$. Is $C$ positive semidefinite ?
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1answer
31 views

Matrix: determinant & Diagonal

There is a question that comes up in my mind after I watched Prof. Gilbert Strang's lectures. He was saying: For any matrix $A$, Since $A = LU$, $\det(A) = \det(LU)$ and $\det(L) = 1$, hence $\det(A) ...
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1answer
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Linear span- geometric interpretation in $\mathbb R^3$

Prove that Span({$x$})={$ax: a\in F$} for any vector $x$ in a vector space. Interpret this result geometrically in $\mathbb R^3$ My attempt at the first part: By definition of Span, Span({$x$}) is ...
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Rank One decrease

Let $X$ be a $N\times N$ real positive semi-definite(p.s.d) matrix with rank $R$. Let $x_1\in Range(X)$ be a non-zero $N\times 1$ vector such that $X_1=X-x_1x_1^T$ is still p.s.d. What is the rank of ...
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Sign of Laplacian in a N-Dimensional Space Under Other Specific Conditions

Consider that $f: \mathbb{R}^N \to \mathbb{R} $ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$: $$ ...
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1answer
81 views

Subsets in $\mathbb{Z}_3$

Let $U_1 \subset (\mathbb{Z}_3)^4$ generated by $(2,1,2,0),(2,0,2,1)$ and $(0,2,0,1)$ and $U_2 \subset (\mathbb{Z}_3)^4$ of equations: $x + y + 2z + 2t = 0$ $x + y + t = 0$ How can I ...
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1answer
32 views

Equation involving a partial trace

Is there, in general, a solution to the following equation? $\text{Tr}_{V_1}(A(X\otimes I_{V_2})) = B$ where A is an operator on $V_1\otimes V_2$, $B$ is an operator on $V_2$, $I_{V_2}$ is the ...
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How to find a linear equation with the same solution set?

I have this homework question that I solved, but it was so easy that I feel like I did something wrong. Can someone just confirm that my approach to this problem was correct? So, I have to find a ...
3
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1answer
53 views

A Question Regards Operators on Inner-Product Spaces, involving $\epsilon$

$V$ is a complex inner product space. Suppose $T\in \mathcal{L}(V)$ is self-adjoint, $\lambda\in F$, and $\varepsilon>0$. Prove that if there exists $v\in V$ such that $\|v\|=1$ and ...
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Computations for LDA: Eigendecomposition

While reading the book Elements of Statistical Learning p. 113, the author used eigendecomposition of the covariance matrix $\hat{\Sigma}_k =\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T$ where ...
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51 views

How to prove a direct sum?

$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint. $i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by ...
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1answer
29 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
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Ortonormal basis of unitary operator and its spectral decomposition - check my solution.

Dear fellow mathematicians, I'm trying to do a linear algebra exercise, but I have no idea whether I have a correct plan of solution. Here is the problem: Find orthonormal eigenbasis (not sure if ...
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1answer
45 views

Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
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2answers
47 views

How to compute dimension of $O(n,\mathbb{R})$

Let $f:GL(n,\mathbb{R})\to GL(n,\mathbb{R})$ be the smooth map $A\mapsto A^TA$. Observe that $f$ has constant rank on $GL(n,\mathbb{R})$ by chain rule and that $O(n,\mathbb{R})$ is the preimage of ...
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1answer
17 views

Vector space subspace of field of polynomial problem

Is the set $W=(f(x)\in P(F): f(x)=0$ or $f(x)$ has degree $n$) a subspace of $P(F)$ if $n\geq 1$? In the answer it is said that no, because addition is not closed. But shouldn't it be closed? If ...
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1answer
26 views

Is a linear transformation just a mathematical description of a straight line?

On Physics Stack Exchange, the question was asked: Are lorentz transformations linear? The up-votes given to an answer seemed to be in proportion to how mathematically sophisticated it was, with mine ...
5
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1answer
46 views

Inequality of Positive-definite matrix.

In this question matrix $A$ is positive-definite if and only if $\forall x\ne0 :x^TAx>0$. ($A$ is not necessarily symmetric) Let $D$ be a positive-definite matrix such that it has block form: ...