Let S equals left curly bracket 1 minus x plus x squared comma 1 plus 2 x comma negative 1 plus 2 x right curly bracket and p with bar on top equals negative 4 plus 8 x minus 4 x squared. The set S is a basis for P subscript 2.
left parenthesis p with bar on top right parenthesis subscript S equals left parenthesis a comma b comma c right parenthesis with
1.2)
Let V be a vector space and v with bar on top element of V. If open parentheses top enclose v close parentheses subscript S equals open parentheses top enclose v close parentheses subscript B for bases S and B of V, then S equals B.
True
False
1.3)
Let V be the set of all ordered pairs of real numbers with addition and scalar multiplication operations defined as follows on V: for top enclose u equals left parenthesis u subscript 1 comma u subscript 2 right parenthesis and top enclose v equals open parentheses v subscript 1 comma v subscript 2 close parentheses in V
top enclose u plus top enclose v equals left parenthesis u subscript 1 comma u subscript 2 right parenthesis plus left parenthesis v subscript 1 comma v subscript 2 right parenthesis equals left parenthesis u subscript 1 plus v subscript 1 minus 1 comma u subscript 2 plus v subscript 2 plus 2 right parenthesis k top enclose u equals k left parenthesis u subscript 1 comma u subscript 2 right parenthesis equals left parenthesis k u subscript 1 minus k plus 1 comma k u subscript 2 plus 2 k minus 2 right parenthesis
This is a vector space.
Explain why the set open curly brackets left parenthesis 1 comma negative 2 right parenthesis comma left parenthesis 1 comma 1 right parenthesis close curly brackets does not form a basis for the space.
Instructions:
Write a sentence explaining why this set is not a basis for V. This question will be marked manually after submission. The final marked achieved for this quiz will therefore only be available after I have gone through all the attempts for this question.