# Tagged Questions

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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### Prove that odd polynomials $f(x)$ of degree $\leq 10$ with $f(-1) = 0$ form a vector space.

Let $P(X)$ be the usual vector space of polynomials in $x$ with real coefficients. Let $U$ denote the subset of $P(X)$ consisting of those elements $f(x)$ which have degree less than or equal to ...
2answers
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### Definition of an angle in a vector space, law of sines

On a lecture in linear algebra we have been given this definition of an angle in vector space with scalar product $\langle , \rangle$: $\cos \alpha=\frac{\langle u,v\rangle}{||u|||v||}$ Throughout ...
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### Show that if $u(t)$ is a unit vector for all $t$ then $u(t)$ and $u'(t)$ are orthogonal for all $t$

Show that if $u(t)$ is a unit vector for all $t$ then $u(t)$ and $u'(t)$ are orthogonal for all $t$. So if $u(t)$ is a unit vector it means that $u(t): \mathbb{R} \to \mathbb{R}$ right? I'm not sure ...
1answer
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### About a finitely generated algebra over a field.

Let $F$ be a field and let $K$ be an associative $F$-algebra which is finitely generated over $F$. Suppose that there exists an element $y ∈ K$ satisfying the condition that for each $v ∈ K$ there ...
4answers
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### Why is an infinite dimensional space so different than a finite dimensional one?

In functional analysis there is a big difference between finite- and infinite-dimensional vector spaces. I have found other questions with nice answers here and here. However, I don't grasp the ...
2answers
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### Find a matrix relative to new bases

Let matrix $T = \left(\begin{array}{cc}1 & 2 \\1 & -1\end{array}\right).$ Let $e_1 = (1,1)^T$ and $e_2 = (3,1)^T$. $T$ is currently relative to the standard basis. If asked to find $T$ ...
1answer
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### Find matrix of linear transformation relative to new bases

If $T:\Bbb R^3\to \Bbb R^2$ is a linear transformation, and the matrix of $T$ = $\left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$. If you use the basis $\{i,j,k\}$ ...
1answer
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### Signature of the norm on $k$-forms.

I wonder about the signature of the inner product on forms on a vector space equipped with a non-degenerate bilinear of signature $(t,s)$. For definiteness, let $(V_{6}, g)$ be an oriented six-...
2answers
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### Calculate new(x,y) for a line

Suppose if tanks has to rotate its main gun by $30^\circ$ to hit the target, what will be its new $(x,y)$ coordinate or a formula to calculate it as shown in image? If bullet is fired from the main ...
0answers
64 views

### Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
3answers
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### Show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0.

Consider $\mathbb{C}$ as $\mathbb{R}$-vector space. If $z,w \in \mathbb{C}$, show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0. I'm just unsure about the question and ...
2answers
453 views

### Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B =${$v_i : i \in I$} and $W_B =${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
1answer
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### Find a basis for the subspace and state the dimension

${\{(a, b, c) : a-3b=0, b-2c=0, 2b-c=0}\}$ The answer states that there is no basis and that the dimension is 0, however I am unsure why. I suspect it is to do with all the equations equalling zero ...
2answers
75 views