Question

For the statements written below, fill in the blanks with numbers that make them logically true statements. Write 'DNE' in cases where there exists no admissible value. Consider the vector space $U = R^1$. Now consider a finite set $A$ such that $A \subset U$. It is impossible for $A$ to be a spanning of $U$ unless it contains at least $\boxed{2}$ distinct elements. Additionally, $A$ must always be a spanning set of $U$ if it has any $\boxed{1}$ distinct elements. I apologize for the inappropriate word choice of the last sentence. Question has been updated to reflect the way I intended to word the question Next, consider the vector space $V = P_n(x)$, the set of polynomials that have a degree of no greater than $n$. Now consider a finite set $B$ such that $B \subset V$. It is impossible for $B$ to be a spanning of $V$ unless it contains at least $\boxed{n+2}$ distinct elements. Additionally, $B$ must always be a spanning set of $V$ if it has any $\boxed{n+1}$ distinct elements. Finally, consider the vector space $W = C[a, b]$, the set of real-valued functions that are defined and continuous on the interval $[a, b]$. Now consider a finite set $C$ such that $C \subset W$. It is impossible for $C$ to be a spanning of $W$ unless it contains at least $\boxed{DNE}$ distinct elements. Additionally, $C$ must always be a spanning set of $V$ if it has any $\boxed{DNE}$ distinct elements.

          For the statements written below, fill in the blanks with numbers that make them logically true statements. Write 'DNE' in cases where
there exists no admissible value.
Consider the vector space $U = R^1$. Now consider a finite set $A$ such that $A \subset U$. It is impossible for $A$ to be a spanning of $U$ unless it
contains at least $\boxed{2}$ distinct elements. Additionally, $A$ must always be a spanning set of $U$ if it has any $\boxed{1}$ distinct elements.
I apologize for the inappropriate word choice of the last sentence. Question has been updated to reflect the way I intended to word the
question
Next, consider the vector space $V = P_n(x)$, the set of polynomials that have a degree of no greater than $n$. Now consider a finite set $B$
such that $B \subset V$. It is impossible for $B$ to be a spanning of $V$ unless it contains at least $\boxed{n+2}$ distinct elements. Additionally, $B$ must
always be a spanning set of $V$ if it has any $\boxed{n+1}$ distinct elements.
Finally, consider the vector space $W = C[a, b]$, the set of real-valued functions that are defined and continuous on the interval $[a, b]$. Now
consider a finite set $C$ such that $C \subset W$. It is impossible for $C$ to be a spanning of $W$ unless it contains at least $\boxed{DNE}$ distinct elements.
Additionally, $C$ must always be a spanning set of $V$ if it has any $\boxed{DNE}$ distinct elements.

for the statements written below fill in the blanks with numbers that make them logically true statements write dne in cases where there exists no admissible value consider the vector space 37499

Added by Rita C.

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