# 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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### Determining a basis for Col($A$) and a dimension for the null space of $A$

Let $A = \begin{bmatrix}1&-1&1&0&-2&1\\1&-1&1&1&0&0\\-1&1&-1&2&5&-1\end{bmatrix}$ a) Determine a basis for Col($A$) b) What is the ...
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### Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
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### What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix}$$ So here the 1st pivot is missing ...
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### Prove that the size and the number of Jordan block's of $\lambda$ and $\bar{\lambda}$ are the same.

Prove that the size and the number of Jordan block's of $\lambda$ and $\bar{\lambda}$ are the same where $T$ is a real operator on $V$ a finite dimensional space. I know that the main is to show that ...
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### Matrices that represent rotations

So the question is What 3 by 3 matrices represent the transformations that a) rotate the x-y plane, then x-z, then y-z through 90? I believe this is the matrix that rotates the xy plane \begin{...
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### Solution for an inequality

I want to solve this inequality for $z$ $$(z+1) \left(1-e^x\right)-e^y\geq 0$$ where $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$. I am struggling because $z$ ...
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### Show that if $d_{i,j} = |i − j|$ then $\sqrt{d}$ is euclidean . [closed]

Is there a standard way to proof that some space is Euclidean? Thanks
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### Convex basis of functions

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$? Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex ...
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### The set of all real or complex invertible matrices is dense

I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, ...
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### Expressing $v$ as a linear combination of $v_1, v_2, v_3$ and Finding $Av$

Let $v_1 \begin{bmatrix}0\\-2\\2\end{bmatrix}, v_2 = \begin{bmatrix}1\\2\\0\end{bmatrix}$ and $v_3 = \begin{bmatrix}2\\0\\-1\end{bmatrix}$ be eigenvectors of the matrix $A$ which correspond to the ...
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### Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
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### Finding formulas for the entries of a matrix

Let $M = \begin{bmatrix}8&2\\-1&5\end{bmatrix}$ Find formulas for the entries of $M^n$ where $n$ is a positive integer $M^n = ?$ (Should be a $2 \times 2$ matrix) What do they mean ...
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### How can I compute $A(v_1 + v_2)$ where $v_1$ and $v_2$ are eigenvectors of the matrix A

If $v_1 = \begin{bmatrix}5\\3\end{bmatrix}$ and $v_2 = \begin{bmatrix}3\\1\end{bmatrix}$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 4$ ...
Southeast Moldings molds plastic handles which cost $\$1.00$per handle to mold. The fixed cost to run the molding machine is$\$3,640$ per week. If the company sells the handles for $\$4.00 $each, ... 1answer 38 views ### finding the solution to$(I_3+A)x=b+2x$I am having trouble solving the$(I_3+A)x=b+2x$without finding the matrix$A$. you are also given the inverse of$A$and the matrix$b$which consist of 3x1. 4answers 55 views ### Writing the solution set(s) of the equation$Ax = 0$Consider the following matrix$A = \begin{bmatrix}1&-4&0&0&1\\0&0&1&0&5\\0&0&0&1&1\end{bmatrix}$a) Write the solution set of the equation$Ax = 0$... 0answers 19 views ### Geometric interpretation of linear programming dual Is there a geometric interpretation of the linear programming dual in terms of the primal? I feel like without some sort of intuition of it, I don't truly understand it. 0answers 32 views ### Dot product of two vectors as the eigenvalue of a special matrix [duplicate] I just noticed that for any two Cartesian vectors their dot product is precisely the only non-zero eigenvalue (if such exists) of the following matrix: $$\vec{a}=(a_1,a_2,a_3,\dots)$$$$\vec{b}=(b_1,... 1answer 15 views ### Identity relative to different orthonormal bases is unitary Let$V$be a finite-dimensional inner product space, and let$\beta,\beta'$both be orthonormal bases for$V$. Is it the case that$[I]^{\beta'}_{\beta}$is unitary? If so, how can we prove this? ... 2answers 31 views ### Can a set of four vectors be a basis for P5? From what I understand, you would need 3 vectors to form a basis of three dimensional space, but does this same restriction apply to a polynomial of let's say P5? In other words, if I'm given W={x^5, ... 2answers 82 views ### Tableau and Simplex Method - No Calculator A non-profit offers crafts complimentary gift packages for its donors. The non-profit costs for each package are \$4 for the Bronze level package, \$7 for the Silver level package, and \$9 for the ...
What are some examples of multiplicative functions on matrices? More precisely, I'm looking for $f : M^{n \times n}, M^{n \times n} \to R$ with the property $f(AB) = f(A)f(B)$ where A, B are \$n \...