# 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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### Linear Algebra -Subspaces

Are the following subsets of $\Bbb R ^n$ subspaces? Justify the answer. a) The set of all vectors $(x_1, x_2, \dots, x_n)$ for which $x_1x_2x_3...x_n = 0$. For this one I worked it out and got that ...
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### Is it possible to have an $n\times n$ real matrix $A$ such that $A^TA$ has an eigenvalue of $-1$?

Question: Is it possible to have an $n\times n$ real matrix $A$ such that $A^TA$ has an eigenvalue of $-1$? I can prove that it is not possible for $n=1,2$, but I am not sure for the general case. ...
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### Linear independence of real powers of x

I know that integer powers of x are linearly independent. I would expect that fractional powers of x (eg. $x^{n/2}$) are also linearly independent. But what about real powers of x? If I could use any ...
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### Quotient group and adjoint matrix

The exercise 1211 in "Problems and Solutions in Mathematics" by Ta-Tsien: Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers. ...
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### An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
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### Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
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### Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y]$ both be two different inner products on V. ...
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### Finding the Jordan Decomposition of a given matrix

Given $$\begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2\\ \end{bmatrix}$$am I need to find a Jordan Decomposition of this matrix. For this purpose I am trying to ...
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### Why more than 3 dimensions in linear algebra?

This might seem a silly question, but I was wondering why mathematicians came out with more than 3 dimensions when studying vector spaces, matrices, etc. I cannot visualise more than 3 dimensions, so ...
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### Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
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### Stationary values

I'm not sure how to go on and answer this question, I appreciate any help. Show that $$f(x) = \ln(3x^2 - 2x -1) - 4x^2$$ has a stationary value when $x$ satisfies $$12x^3 - 8x^2 - 7x + 1 =0$$
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### operator norm of a matrix and the largest eigenvalue

Is it true that for any $n \times n$ matrix $A$, with real entries, and where all eigenvalues are real, and non zero, the operator norm ( where$||A|| = \max_{|x|=1} |Ax|$ with |.| is the standard ...
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### Find a function $h$ such that $g(x) =\langle f, h \rangle$

Let $P_2(\mathbb R)$ be an inner product space with $\langle f, h \rangle = \int_{0}^{1}f(t)h(t)dt$. Let $g(f) = f(0) + f'(1)$. Find $h(t)$ such that $g(f) = \langle f,h \rangle$. I tried ...
Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bounded linear operator. Then how to show the following? The range of $T$ is finite-dimensional if and only if $T$ can be represented in ...
### Polynomial in several variables over $GF(2)$
Can anyone please explain how this Lemma has been proved? Lemma: Let $f$ be a nonzero polynomial in variables $x_1,\ldots,x_n$ over $GF(2)$, and let $d$ be the maximum degree of $f$ with respect to ...