# 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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### Find subspace $T$ of space $\mathbb R^3$ so that $\mathbb R^3=S \oplus T$

I have one problem. I am sure it is not complicated, but I only need help to see am I, at least, on the right path. Problem: Let $S=Span\{(0,-2,3),(1,1,1),(2, -2, 8)\}\subseteq \mathbb R^3$. Find ...
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### Sylvester's argument for bilinear functions

Let $V$ be a vector space of dimension $n$ and let $b:\colon V \times V\to \mathbb{R}$ be a symmetric bilinear function. Sylvester's theorem says that there exists a basis of $V$ with respect to ...
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### Basis for the space of linear transformations $L(\Bbb R^3,\Bbb R_3[x])$

How do I build a basis for the vector space $L(\Bbb R^3,\Bbb R_3[x])$? This is the vector space of all linear transformations that goes from $\Bbb R^3$ to the space of polynomials of degree 3 ...
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### Wedge product is nondegenerate symmetric bilinear form

Let$$f: \Lambda^k(\mathbb{R}^n) \times \Lambda^{n - k}(\mathbb{R}^n) \to \mathbb{R}, \quad f(\alpha, \beta) = \alpha \wedge \beta.$$How do I see that $f$ is a nondegenerate symmetric bilinear form?
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### vector generation by linear combination

I have 4 vectors in $R^3$ given as: $v1=(-1,2,0), v2=(3,1,2), v3=(4,-1,0), v4=(0,1,-1)$. I have to show that the vector $v= (5,6,0)$ can be generated by a linear combination of this vector. let the ...
Suppose I have the vectors $\alpha, \beta \in \mathbb{R}^2$ with inner products $(\alpha, \alpha) = 1$ and $(\beta, \beta) = 2$, and the angle between $\alpha$ and $\beta$ is $\theta = \frac{3\pi}{4}$....
what is the directional derivative of$f(x,y)=xy+x^2$ at the point $(2,-1,1)$ in the direction $(1,3,-1)$? So the unit vector is $\frac{(1,3,-1)}{\sqrt{11}}$, now we have to take the gradient of ...