SB True/False Exercises

Exercises 3.8 True/False Exercises

View Source for exercises

<exercises xml:id="true-false-exercises">
  <title>True/False Exercises</title>
  <exercise xml:id="true-false-one" label="vector-space-dimension">
    <title>True/False</title>
    <idx>vector space</idx>
    <statement correct="no">
      <p>
        Every vector space has finite dimension.
      </p>
    </statement>
    <feedback>
      <p>
        The vector space of all polynomials with finite degree has a basis,
        <m>B = \{1,x,x^2,x^3,\dots\}</m>, which is infinte.
      </p>
    </feedback>
    <hint>
      <p>
        <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
        has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
        content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
      </p>
    </hint>
  </exercise>
</exercises>

1. True/False.

View Source for exercise

<exercise xml:id="true-false-one" label="vector-space-dimension">
  <title>True/False</title>
  <idx>vector space</idx>
  <statement correct="no">
    <p>
      Every vector space has finite dimension.
    </p>
  </statement>
  <feedback>
    <p>
      The vector space of all polynomials with finite degree has a basis,
      <m>B = \{1,x,x^2,x^3,\dots\}</m>, which is infinte.
    </p>
  </feedback>
  <hint>
    <p>
      <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
      has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
      content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
    </p>
  </hint>
</exercise>

Every vector space has finite dimension.

True.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
False.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.

Hint.

View Source for hint

<hint>
  <p>
    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
    has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
    content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
  </p>
</hint>

\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)