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.

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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 \(n\text{?}\)

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