# Tagged Questions

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### Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
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### Does there exist a multiplication operator so that the set of all rational over irrationals is a vector space? [closed]

I am trying to prove whether the set of rationals over the irrational field can be a vector space.
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### If $\ker(t) = 0$, then is $t$ a linear transformation? [closed]

Suppose there is a transformation $\mathbb R^n \to \mathbb R^m$, with $0 < m < n$, and they ask whether it is linear. It says that since $n - m > 0$, the kernel is non-zero, which makes it ...
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### Do polynomials $P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
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### Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
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### groups products vs vector space products

I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted ...
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### Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring $$S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.$$ Show ...
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### How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
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### Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
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### What is the difference between $\mathcal{X}\subseteq\mathbb{R}^n$ and $\mathcal{X}\subset\mathbb{R}^n$

Let $\mathcal{X}$ be a non-empty set. For instance, let it be a set of vectors of the form $\mathbf{x}\in\mathbb{R}^n$, i.e., ...
### Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$
Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...