# Tagged Questions

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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### Can a Matrix with positive entries have a negative eigenvalue?

It seems intuitive to me that the answer is no, but I can't prove it.
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### existence of $\lambda$ in $V^*$ in an $n$ dimensional vector space

Let $V$ be an $n$-dimensional vector space, and $U \subset V$ a subspace of dimension $n−1$. 1) Show that there exists $λ∈V^∗$ with $\ker(λ)=U$. 2) Show that if $μ ∈ V^∗$ is another linear ...
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### Derivative of $f(A) = \|A x\|^2$ with respect to the Matrix

Suppose I have $A \in \mathbb{R}^{n^2}$ and $x \in \mathbb{R}^n$ where $A$ is interpreted as a matrix. We can define $f(A) = ||A x||^2$ for some constant $x$. What is the derivative of $f$, written ...
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### Find set of solutions $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $z\in\mathbb{R}^N$.

How to characterize $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $z\in\mathbb{R}^N$? Is there also a general way for more complex equations $y'\beta(z)=\iota'z$ where $\beta(z)\in\mathbb{R}^N$ ...
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### Geometric interpretation of Linear Independence and absolute value

I am wondering something in regard to linear independence / dependence of functions. I understand when working with vectors in $\mathbb{R^n}$ for example, we can make the geometric link that vectors ...
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### Linear Algebra - Linear Combinations

We are asked the following: The span of $$v_1,v_2,…,v_r$$ can be the whole $$V,+,⋅$$ or a subset of $$V,+,⋅.$$ Give an example of each situation. The span is denoted $$span v_1, v_2, ..., v_r$$ ...
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### Can the terms in a 3x3 determinant be any six nonzero numbers?

Given six nonzero real numbers $x_1,\ldots x_6$, can you construct a 3x3 matrix such that the six diagonal products that appear in the determinant are $x_1,\ldots,x_6$, respectively? In other words, ...
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### suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,…$. why $A$ and $B$ are same characteristic polynomial? . [duplicate]

Let $A,B \in {M_n}$ and suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,...$ . Why do $A$ and $B$ possess the same characteristic polynomial?
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### Find values of $t$ so that this matrix is positive definite

I will start from this point: $\det{\left(B-\lambda I\right)}=0\Longleftrightarrow\begin{vmatrix}t-\lambda&3&1\\3&t-\lambda&0\\1&0&t-\lambda\end{vmatrix}=0$ Now we will ...
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### Zero characteristic polynomial?

Is it possible that characteristic polynomial of an $n \times n$ matrix be the zero polynomial? If this happens, this means that any scalar would serve as an eigenvalue?
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### Problem on eigenvalues

Let A be real square matrix of order $n \geq 2$. Then show that: A. if $A^3 - I$ is singular, then $1$ is eigenvalue of $A$ B. if $A$ is singular, then $I+2A+A^2$ has eigenvalue $1$ My ...
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### Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme $X$...
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### How is the perturbation in one column of a symmetric matrix reflected in its eigenvalues?

Suppose we have a 0-1 square symmetric matrix. Then it's eigenvalues are real. But I have observed that by multiplying any of its column by a positive constant, even if the matrix is not symmetric ...
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### Why : if the matrix $A$ is not invertible, then $L_A$ is not onto.

I was reading a book and the following statement was made: If $A$ is not invertible, then $L_A$ is not onto. Here, the matrix $A$ is $n \times n$ I'm just curious as to why this is true. Thank you!...
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### addition of two vectors in $\mathbb R^2$

Confused about this question ..vectors in space $\mathbb{R}^2$ $$M = \{(x,y)∈ \mathbb{R}^2 | \ 0 ≤ x ≤y\}$$ $$N = \{(x,y)∈ \mathbb{R}^2 | \ 0 ≥ x ≥y\}$$ $$P= M∪N$$ ($P$ is closed under scalar ...
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### Prove that $A$ diagonalizable.

Let $A$ be an $n \times n$ matrix, and let $v_1,...,v_n$ be a basis of $R^n$ so that each $v_i$ is an eigenvector of $A$. Prove that $A$ diagonalizable. Does the diagonalization of $A = QDQ^{-1}$ ...
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Suppose A is similar to the matrix B given below. $$B= \begin{bmatrix} 7 & 0 & 0 \\ a_{21} & 4 & 0 \\ a_{31} & a_{32} & -0.5 \\ \end{bmatrix} ... 1answer 50 views ### Does multiplication by the inverse of a Cholesky matrix preserve order? Let n \in \{1, 2, \dots\} and let C \in \mathbb{R}_{n, n} be a real, symmetric and positive definite n \times n matrix. Define B \in \mathbb{R}_{n, n} to be the real, lower triangular matrix ... 1answer 31 views ### Checking vectors for subspaces in \mathbb{R}^3 space. How to check if these sets are subspaces in \mathbb{R}^3 ? i know the three condtions but how to check those conditions with some solvings? Thanks in advance......$$U_1 = \{(x,y,xy)\mid x,y ∈ \...
I am confused of solving expnential functions they look easy but cant solve it. 1: $$\large e^{8\cdot\ln(b^{1/4})}$$ and this one solving for x: 1: $$\ln(6x-2) = 5$$ FYI : Its not an assignment. i ...