find a base to U Linear Algebra

Question

dear users please help me... im answering a long question now ive been guided to find a base to U at the end of the process i got this $u= Sp\{x^4-3x^3+2x^2, 3x^4-7x^3+4x ,1\}$ and ive been guided to check by converting to R5, that those 3 vectors which span U are not linearly independent and because that U is a base and . dimU = 3.

i didnt understood how i do this process... thank you very much and im new so sorry if you didnt understood

Answer 1

You can identify the set of polynomials of degree at most four with $\mathbb{R}^5$ in the following way:

to any polynomial $ax^4+bx^3+cx^2+dx+e$ assign the vector $(a,b,c,d,e)\in\mathbb{R}^5.$

So, you can think of $U$ as spanned by the vectors

$$(1,-3,2,0,0),(3,-7,0,4,0),(0,0,0,0,1)\in\mathbb{R}^5.$$

These vectors are linearly independent, that is,

$$x(1,-3,2,0,0)+y(3,-7,0,4,0)+z(0,0,0,0,1)=(0,0,0,0,0)\Rightarrow x=y=z=0.$$ Thus they constitute a basis for $U,$ from where, it follows that $\text{dim}(U)=3.$

Answer 2

It seems that what you're asking is how to convert these polynomials to coordinate vectors relative to the standard basis of the vector space of $4$th degree polynomials.

What it comes down to is the following: the standard basis of (4th degree) polynomials is given by $$ \mathcal{B} = \{1,x,x^2,x^3,x^4\} $$ For convenience, I'm going to define $e_1 = 1, e_2 = x,\dots, e_5 = x^4$. These polynomials $e_i$ are our basis vectors. With this in mind, we can write $$ x^4 - 3x^3 + 2x^2 = 0e_1 + 0e_2 + 2e_3 - 3e_4 + 1e_5 = \begin{bmatrix} 0\\0\\2\\-3\\1 \end{bmatrix}_{\mathcal B} $$ In such a fashion, we may state that $$ U = span\left\{ \begin{bmatrix} 0\\0\\2\\-3\\1 \end{bmatrix}_{\mathcal B}, \begin{bmatrix} 0\\4\\0\\-7\\3 \end{bmatrix}_{\mathcal B}, \begin{bmatrix} 1\\0\\0\\0\\0 \end{bmatrix}_{\mathcal B} \right\} $$

Answer 3

We have a natural identification of $\Bbb R_4[x]$: the linear space of polynomials with degree at most equal $4$ and $\Bbb R^5$ in this way $$a_0+a_1x+a_2x^2+a_3x^3+a_4x^4\leftrightarrow (a_0,\ldots,a_4)$$ hence $U$ is a linearly independent iff the vectors $(1,0,0,0,0)$, $(0,4,0-7,3)$ and $(0,0,2,-3,1)$ are linearly independent in $\Bbb R_4[x]$.