Discrete Math and Vector Spaces
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Questions and Answers

What can be said about the set L(A) of all linear combinations of elements of a set A?

  • It is a subspace. (correct)
  • It is not a subspace.
  • It is only closed under addition.
  • It is not closed under addition and multiplication by scalars.

    • Why does L(A) include each ai?

  • Because L(A) contains all linear combinations of elements of A. (correct)
  • Because L(A) is closed under addition.
  • Because L(A) is closed under multiplication by scalars.
  • Because L(A) is a subspace.

    • What can be said about a subspace W that includes A?

  • It contains each linear combination L Xiai. (correct)
  • It does not contain each linear combination L Xiai.
  • It cannot contain any linear combinations L Xiai.
  • It only contains some linear combinations L Xiai.

    • How can a sum of two linear combinations of elements of A be rewritten?

    <p>As a finite sum of scalars times elements of A. (D)</p> Signup and view all the answers

    What does it mean for a set A to span a vector space V?

    <p>L(A) is equal to V. (C)</p> Signup and view all the answers

    Is the function space on an infinite set A always finite-dimensional?

    <p>No (A)</p> Signup and view all the answers

    What is the space of all mappings f: A -> W?

    <p>A vector space of functions (D)</p> Signup and view all the answers

    What is Hom(V, W)?

    <p>A subset of WV consisting of all linear mappings (C)</p> Signup and view all the answers

    What is the theorem 3.2 stating?

    <p>Hom(V, W) is a vector subspace of WV (C)</p> Signup and view all the answers

    Why is Hom(V, W) a subspace?

    <p>Because it is closed under addition and scalar multiplication (C)</p> Signup and view all the answers

    What is the theorem 3.3 stating?

    <p>The composition of linear maps is linear (C)</p> Signup and view all the answers

    What is the result of the composition of linear maps?

    <p>A linear mapping from V to X (C)</p> Signup and view all the answers

    What is the value of L~ at t, given the coefficients xi and functions Ii?

    <p>Xl sin t + X2 cos t + X3et (B)</p> Signup and view all the answers

    What is the image of ~1 under the linear mapping T?

    <p>sin t (C)</p> Signup and view all the answers

    What is the set of all real-valued functions on IR?

    <p>IRR (A)</p> Signup and view all the answers

    What is the theorem that states that every linear mapping from IRn to W is of a certain form?

    <p>Theorem 1.2 (D)</p> Signup and view all the answers

    What is the coefficient of et in the linear combination L~?

    <p>X3 (D)</p> Signup and view all the answers

    What is the spanning set for IRn?

    <p>{~i} (C)</p> Signup and view all the answers

    What is the property of the linear combination map T?

    <p>Linearity (B)</p> Signup and view all the answers

    What is the result of T(x + y) in terms of T(x) and T(y)?

    <p>T(x + y) = T(x) + T(y) (C)</p> Signup and view all the answers

    What is the significance of the theorem in this context?

    <p>It is a tremendously important theorem (C)</p> Signup and view all the answers

    What is the notation La for?

    <p>A linear combination mapping from IR to W (A)</p> Signup and view all the answers

    What is the skeleton of a linear mapping T?

    <p>The n-tuple {T(5i)} (C)</p> Signup and view all the answers

    What is the relationship between the map a -> La and the set of all linear maps T from IR to W?

    <p>They are bijective (C)</p> Signup and view all the answers

    What is the topic of Chapter 3 in the table of contents?

    <p>Vector Spaces (B)</p> Signup and view all the answers

    What is the title of the chapter that deals with 'Bases Dimension The dual space'?

    <p>Chapter 3 (A)</p> Signup and view all the answers

    What comes after 'Chapter 3' in the table of contents?

    <p>Chapter 4 (A)</p> Signup and view all the answers

    What is the topic of the chapter marked with a *?

    <p>The Differential Calculus (B)</p> Signup and view all the answers

    How many chapters are there in the table of contents?

    <p>6 (A)</p> Signup and view all the answers

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