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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Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but it's ...
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Linear programming algorithm that minimizes number of non-zero variables?

I have real world problems I'm trying to programmatically solve in the form of $$Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ Subject to \begin{align} & a_{11} x_1 + a_{21} x_2 + \cdots + a_{n1} ...
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Square block matrix, with Hermitian, non-negative definite blocks, prove that the matrix is also non-negative definite,

Consider the square block matrix $$S= \begin{bmatrix} R & RQ^* \\ QR & QRQ^* \\ \end{bmatrix} $$ where $R$ is a Hermitian, non-negative definite square matrix ...
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1answer
49 views

I can't understand a step in the proof of the associativity of matrix multiplication

Matrix multiplication is proven by the following reasoning: Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then $$ ...
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24 views

Over-specified linear system

Consider the matrix $A $ with RREF consisting of three of the 4, 4- dimensional standard vectors: $[\mathbb {e_1}, \mathbb {e_2}, \mathbb {e_3} ] $ Since the rank is 3 the matrix has one solution ...
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61 views

Prerequisites for Linear Algebra Done Right by Sheldon Axler.

I've read some notes online and I learned so far: $\{\overset{\displaystyle\ldots}\ldots$ Systems of Two Linear Equations ...
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23 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
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41 views

Linear independence related with functions

Good day ! I don't understand the following problem: "Prove that the three functions $x^2,\cos{x},e^x$ are linearly independent" So I think so I have to prove that the linear combination: $a ...
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78 views

What does this theorem in linear algebra actually mean?

I've just began the study of linear transformations, and I'm still trying to grasp the concepts fully. One theorem in my textbook is as follows: Let $V$ and $W$ be vector spaces over $F$, and ...
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27 views

Subset of $\mathbb{R}^4$ such that the intersection with an hyperplane is dense and does not contains $4$ coplanar points.

Does it exist a subset $S$ of $\mathbb{R}^4$ such that for all affine hyperplane $H\subset \mathbb{R}^4$, the set $H \cap S$ is dense in $H$ and does not contains $4$ coplanar points? More than ...
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28 views

how to find the dimension of the image of $f$ in this case?

Let $A \in M_{m \times n}(\Bbb R)$ be fixed, and let $B \in M_{m \times l} (\Bbb R)$. Consider the map $f: M_{n \times l}(\Bbb R) \to M_{m \times l}(\Bbb R)$ defined by $f(X) = AX + B$ for all ...
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21 views

Determining the standard matrix from the images of the standard basis vectors

Let a linear transformation $T:$ $\mathbb{R}^3$ → $\mathbb{R}^3$ rotate a vector around the z-axis by $45^{o}$ followed by an orthogonal projection onto the x-axis. Determine the standard matrix ...
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2answers
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Basis for kernel of trace map

Let $T: M_{n \times n}(F) \rightarrow F$ be defined by $T(A) = tr(A)$. I want to find out what a basis is for the kernel $N(T)$ of this linear map. I know $tr(A) = \sum_{i=1}^{n} A_{ii}$. I also ...
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1answer
62 views

Linear independence question (do 2 vectors who are not multiples of one another and a third which is not in their span form R^3?

The True/False question is: Suppose that $v_1, v_2, v_3$ are in $\Bbb R^5$, $v_2$ is not a multiple of $v_1$, and $v_3$ is not a linear combination of $v_1$ and $v_2$. Then $\{v_1, v_2, v_3\}$ is ...
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24 views

Algebra Problem: Two airplanes flying in opposite directions.

I tried to create the equation and failed. Maybe somebody could help me? Two airplanes start flying from an airport in opposite directions, one averaging a speed of 40 km/hr greater than that of the ...
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Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
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Finding bases for kernel N(T) and image R(T)?

Goodday, I need some help with the following problem: Find bases for both N(T) and R(T) in the following transformation: $T: M_{2 \times 3}(F) \rightarrow M_{2 \times 2}(F)$ defined by: ...
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Find the solution of linear equation using Wiedemann/ Krylov method

I am using Wiedemann (some literature called Krylov method) to find the solution of a linear equation that defined as $$Mx=b$$ Instead of resolving entire elements of x (size $K \times 1$), we can ...
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1answer
22 views

Linear ALgebra Subspaces over sets

Let $V$ be a vector space over a field $F$ and $M,N ≤ V$. Consider the following subsets of $V$: (a) $M \cup N$ (b) $M \cap N$ (c) $M+N$ (d) $M - N$ For each of the subsets in (a)-(d) above, ...
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18 views

Inverse of nonnegative Toeplitz matrice

Consider a right-hand circulant matrice of size $n$ (called also Toeplitz matrice) \begin{equation} T= \left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_n \\ a_n & a_1 & ...
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1answer
44 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
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Prove that M-matrix is invertible? [on hold]

Prove that a matrix $A = (a_{ij})$, of size $p × p$ verifies the following conditions $$(a_{ij})\leq 0$$ for all $i \not= j$ and $$\sum_{j=1}^{p}(a_{ij})> 0$$ for all $i$ is invertble.
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expressing canonical base of univariate polynomials in binomial base

Two bases are fairly standard for ${\mathbb Q}[X]$ : the canonical base $(X^j)_{j\geq 0}$ and the binomial base $(b_j(X))_{j\geq 0}$ where $b_j(X)=\binom{X}{j}=\frac{X(X-1)\ldots (X-(j-1))}{j!}$ (thus ...
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3answers
53 views

Properties of eigenvalues [on hold]

Let $A\in M_n(\mathbb{R})$ be a diagonalizable matrix over $\mathbb{R}$ Prove that there exist $\lambda\in \mathbb{R}$ such that every eigenvalue of $A+\lambda I_n$ is positive.
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1answer
29 views

stuff involving adjoint, self adjoint [on hold]

Let $T: V \to V$ be a linear transformation relative to a finite dimensional Euclidean space $V$ (real or complex). Prove that there exists linear transformation $T^*: V \to V$ (called the adjoint ...
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55 views

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation

If $A$ is diagonalizable, using the Bauer-Fike theorem, for any eigenvalue $λ$ of $A$, there exists an eigenvalue $μ$ of $A+E$ such that $|\lambda-\mu|\leq\|E\|_2$ (the vector induced norm). Here I ...
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4answers
74 views

$A.A^t$ is diagonal

Be $A$ a semidefinite nonnegative matrix. What kind of conclusions can we say about $A$ if $A.A^t$ is diagonal? Same question when $A$ is binary matrix. Thanks
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1answer
42 views

Künneth formula in topology, show isomorphism

Where could I find a proof of the isomorphism aspect of Theorem 2.4 in this pdf: http://math.stanford.edu/~conrad/diffgeomPage/handouts/tensor.pdf For vector spaces $V$ and $W$, consider $V$ and ...
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1answer
60 views

How to show trace of $AB$ is zero for $A \in \mathfrak{u}_n$ and $B \in \mathcal{H}_n$?

Please have a look at this question: Help needed in understanding the basics of Cartan decomposition of a Lie algebra I want to show that the decomposition $\mathfrak{gl}_n = u_n \oplus ...
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1answer
60 views

If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$ then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$

The given question is If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$, then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$. Any hints?
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If $A$ is Hermitian and $B$ is skew-hermitian matrix, then trace of $AB$ is zero?

I reduced a problem to showing that if $A \in End (\mathbb{C}^n)$ is such that $A=-\bar{A}^t$ and if $B\in End (\mathbb{C}^n)$ such that $B=\bar{B}^t$ then the trace $Tr(AB)=0$. But I cannot show ...
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1answer
81 views

Is Hoffman-Kunze a good book to read next?

I'm planning on self-studying linear algebra, and trying to decide on a book. I'm thinking of using Hoffman and Kunze. What sort of experience is required to handle Hoffman and Kunze? So far, I've ...
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1answer
50 views

Finding the eigenvalues and eigenvectors of $A^{n}$

Find the eigenvalues and eigenvectors of $A^{5}$ for $A = \begin{bmatrix} 0&0&-1 \\-1&1&-1 \\ 1&-1&0\end{bmatrix}$. How many eigenspaces does it have? What is the dimension ...
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1answer
31 views

show $\int_0^1 f(t)g(t) dt$ is a non-degenerate scalar product; $f,g \in C[0,1]$

Let $C[0,1]$ be the vector space of continuous real-valued functions on the interval [0,1]. The following mapping is defined: $$ \langle \bullet , \bullet \rangle : C[0,1]^2 \rightarrow \mathbb{R} : ...
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1answer
58 views

Self-adjoint matrices: prove that $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$ [on hold]

$A,B \in M_n(\mathbb C)$ and self-adjoint. Prove the following inequality: $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$. Thanks
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2answers
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linear map $f:V\rightarrow V^*$ or $\mathbb F$.

I'm having a bit of trouble understanding the dual space $V^*$ to a vector space $V$ over field $\mathbb F$. So far I understand that a linear form/functional $f$ is a linear map from $V$ to its ...
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Finding all matrices for which the homogeneous system has a given solution space

Find all $3\times 3$ matrices for which the homogeneous system has a solution space as the line $x = 2t$, $y = t$, $z = 0$. (Hint: Write the row reduced augmented matrix from given information.) ...
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How to divide two polynomials using point-value representation

I'm wondering whether there is any way to divide two polynomials represented in point-value forms ? Or Is there any tricks I can use? Point-value representation: A polynomial $f$ is evaluated at ...
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2answers
58 views

Given $A$ is $6×6 $ real symetric matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $

Given $A$ is $6×6 $matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $. I knowthat rank of matrix doesnot change when we square it , but how to proceed in this question.Any hints ? Thanks
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Basis if and only if $\varphi$ is an isomorphism

Let $V$ be a finite dimensional vector space. I am aware of the theorem stating that if $\varphi \colon V \rightarrow V$ is an automorphism and $\mathcal{A}=(a_1, \ldots, a_n)$ is a basis of V then ...
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Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
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1answer
31 views

What do we know about the solution of this set of linear equations?

Let $C \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, $Y \in \mathcal{Y}$, and $B : \mathcal{Y} \to \mathbb{R}^n$ be linear, where the linear space $\mathcal{Y} \subset \mathbb{R}^m$ may be ...
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2answers
73 views

Cube roots escape [on hold]

$ \sqrt{\sqrt[3]{5}-\sqrt[3]{4}} \times 3 = \sqrt[3]{a} + \sqrt[3]{b} - \sqrt[3]{c}, $ where $ a, b $ and $ c $ are positive integers. What is the value of $ a+b+c $? This question appeared in one ...
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Request a paper by Gelfand and Ponomarev

I am looking for the following paper by Gelfand & Ponomarev: I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a ...
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1answer
20 views

What this vector equation means

I have this equation K = (A, B, C) / |A x B| A, B, C are 3D vectors. K should be number, not vector. How is this calculated, or what this expression means?
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Multiplications of non-square matrices and the dependencies of row vectors.

I'd like to find $D$ and $L$ for a given $H$. $H$ is a 7-by-6 matrix. Its rank is 6. All sub-matrices of $H$ are full rank. In other words, if we choose any $n$-by-$n$ sub-matrix within $H$, where $n ...
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1answer
22 views

Basis for the space of quadratic polynomials orthogonal to those with $p(2)=p(1)$

Let $P_2[x]$ be the space of polynomials of degree less than or equal to 2. If $W = \{p ∈ P_2[x] \mid p(2) = p(1)\}$, then find a basis for $ W^⊥$ where $P_2[x]$ is equipped with an inner product ...
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11 views

Optimum sampling of equation

While solving a ill conditioned under determined linear system Ax = b, I have huge number of equations(lets say 10^4) or say relationship between unknowns. I don't want to use all relationships to ...
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1answer
27 views

How do row operations affect the column space?

I've been curious about this: Row operations do not affect the row space, but they affect the column space. Is there any way to 'systematically' perform row operations to make the column space the ...
4
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1answer
63 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?