A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors
Problem 150
Show that the set
\[S=\{1, 1-x, 3+4x+x^2\}\]
is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.
Show that the set
\[S=\{1, 1-x, 3+4x+x^2\}\]
is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.
Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let
\[U=\{ p(x) \in P_n(\R) \mid p(1)=0\}\]
be a subspace of $P_n(\R)$.
Find a basis for $U$ and determine the dimension of $U$.
Add to solve laterLet $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Add to solve later