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Quiz: Vector Spaces and Inner Products

Test your understanding of abstract vector spaces and inner product concepts.

1. A set \(V\) is a vector space if it satisfies:

  1. Contains only numerical vectors
  2. Is closed under vector addition and scalar multiplication with 8 axioms satisfied
  3. Contains exactly three vectors
  4. Has a unique basis
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The correct answer is B. A vector space must be closed under vector addition and scalar multiplication, and satisfy eight axioms including associativity, commutativity of addition, existence of zero vector, and distributive laws.

Concept Tested: Vector Space Axioms

2. A subspace of a vector space must:

  1. Have the same dimension as the parent space
  2. Contain the zero vector and be closed under addition and scalar multiplication
  3. Be finite-dimensional
  4. Contain at least two vectors
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The correct answer is B. A subspace must contain the zero vector and be closed under vector addition and scalar multiplication. These conditions ensure the subspace is itself a vector space under the inherited operations.

Concept Tested: Subspace

3. An inner product \(\langle \mathbf{u}, \mathbf{v} \rangle\) must satisfy all EXCEPT:

  1. Linearity in the first argument
  2. Conjugate symmetry
  3. Positive definiteness
  4. Multiplicativity: \(\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{u} \rangle \cdot \langle \mathbf{v} \rangle\)
Show Answer

The correct answer is D. Inner products require linearity (in first argument for real, conjugate-linear in second for complex), conjugate symmetry \(\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}\), and positive definiteness. There is no multiplicativity requirement.

Concept Tested: Inner Product Properties

4. The standard inner product in \(\mathbb{R}^n\) is:

  1. The sum of vector components
  2. The dot product \(\mathbf{u} \cdot \mathbf{v} = \sum u_i v_i\)
  3. The cross product
  4. The Euclidean distance
Show Answer

The correct answer is B. The standard inner product in \(\mathbb{R}^n\) is the dot product: \(\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i\). It induces the Euclidean norm.

Concept Tested: Standard Inner Product

5. An orthonormal basis has vectors that are:

  1. Parallel and of any length
  2. Mutually orthogonal and each with unit length
  3. Linearly dependent
  4. All equal to each other
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The correct answer is B. An orthonormal basis consists of vectors that are mutually orthogonal (pairwise dot product is zero) and each normalized to unit length. This makes coordinate representation and computation especially convenient.

Concept Tested: Orthonormal Basis

6. The projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) is:

  1. \(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}\)
  2. \(\mathbf{u} + \mathbf{v}\)
  3. \(\mathbf{u} \times \mathbf{v}\)
  4. \(\|\mathbf{u}\| \|\mathbf{v}\|\)
Show Answer

The correct answer is A. The orthogonal projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is \(\text{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}\). This gives the component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\).

Concept Tested: Orthogonal Projection

7. The orthogonal complement of a subspace \(W\) contains:

  1. All vectors parallel to \(W\)
  2. All vectors orthogonal to every vector in \(W\)
  3. Only the zero vector
  4. The basis vectors of \(W\)
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The correct answer is B. The orthogonal complement \(W^\perp\) consists of all vectors that are orthogonal to every vector in \(W\). For any \(\mathbf{u} \in W^\perp\) and \(\mathbf{w} \in W\), we have \(\langle \mathbf{u}, \mathbf{w} \rangle = 0\).

Concept Tested: Orthogonal Complement

8. The Cauchy-Schwarz inequality states:

  1. \(|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \|\mathbf{v}\|\)
  2. \(\|\mathbf{u} + \mathbf{v}\| = \|\mathbf{u}\| + \|\mathbf{v}\|\)
  3. \(\langle \mathbf{u}, \mathbf{v} \rangle = \|\mathbf{u}\|^2\)
  4. \(\|\mathbf{u} - \mathbf{v}\| > \|\mathbf{u}\|\)
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The correct answer is A. The Cauchy-Schwarz inequality bounds the inner product: \(|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \|\mathbf{v}\|\). Equality holds if and only if the vectors are parallel.

Concept Tested: Cauchy-Schwarz Inequality

9. An inner product space is:

  1. A vector space with a defined inner product
  2. A space containing only unit vectors
  3. A subspace of \(\mathbb{R}^3\)
  4. A space with no zero vector
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The correct answer is A. An inner product space is a vector space equipped with an inner product function that satisfies the required axioms. This structure enables defining lengths, angles, and orthogonality.

Concept Tested: Inner Product Space

10. The angle \(\theta\) between two non-zero vectors is determined by:

  1. \(\cos\theta = \|\mathbf{u}\| + \|\mathbf{v}\|\)
  2. \(\cos\theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}\)
  3. \(\theta = \mathbf{u} \cdot \mathbf{v}\)
  4. \(\sin\theta = \frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}\)
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The correct answer is B. The angle between vectors is given by \(\cos\theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}\). This generalizes the geometric definition of angle to any inner product space.

Concept Tested: Angle Between Vectors