TITLE: Writing Vectors As A Linear Combination Of Other Vectors
QUESTION [0 upvotes]: I've been tangled with a section from a question :
In the vector space of  $P_3(R)$ given a group of vectors :
$$S= \{u_1=1+t+t^2 , u_2=2+3t+3t^2 , u_3=2+6t-t^2 , u_4=1-2t+2t^2+5t^3\}$$
I have managed to check whether they are linearly independent or not and found the basis and the dimension of $S$
$$B_S=\{(1,0,0,1),(0,1,0,1),(0,0,1,-1)\}$$
$${\operatorname{dim}_S=3}$$
and what is left is to type all the vectors that belong to S as a combination for the basis i found , kind of stuck there , any suggestions ?

REPLY [0 votes]: The set $S$ is equal to$$\{a+bt+ct^2+dt^3\mid a+b=c+d\}.$$This is a $3$-dimensional vector subspace of, which contains $1+t^3$, $t+t^3$, and $t^2-t^3$, which are the elements of your basis.
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