Linear Algebra: Orthogonal Sets
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Questions and Answers

What characterizes the set Y as an orthogonal basis for W?

  • All vectors in Y are of equal length.
  • Y spans the entire space W. (correct)
  • The vectors in Y are linearly dependent.
  • The dot product of any two distinct vectors in Y is zero. (correct)

    • In the expression Spandv,..., v3 Spanxi, what does 'Span' refer to?

  • The dimension of the vector space.
  • The set of linear combinations of a given set of vectors. (correct)
  • A specific vector in the basis.
  • A unique point in the vector space.

    • What does it imply if W = Spandx xn3?

  • W has a dimension greater than 3.
  • W consists of all linear combinations of the vector x. (correct)
  • W is defined without reference to the vector x.
  • W is specifically spanned by a single vector x.

    • What is the outcome of constructing an orthogonal basis Ev r23 for W?

    <p>Provides a simplification in calculations within W.</p> Signup and view all the answers

    When referring to the expression v = (2), what does the number 2 represent?

    <p>A specific component of the vector.</p> Signup and view all the answers

    What condition must be met for a basis of W to be considered orthogonal?

    <p>The dot product of any two distinct basis vectors must be zero.</p> Signup and view all the answers

    In the context of orthogonal bases, which expression properly represents the summation of vectors?

    <p>$y = Up1 + Up2 + Up3$</p> Signup and view all the answers

    Which interpretation can be drawn from the statement involving E and the orthogonal basis?

    <p>E can be any vector in the space defined by the basis.</p> Signup and view all the answers

    If a basis is orthogonal, what can be said about the angles between the basis vectors?

    <p>The angles between basis vectors are all right angles.</p> Signup and view all the answers

    Which of the following is true for an orthogonal basis concerning vector components?

    <p>Each vector is solely represented by its projection onto the basis vectors.</p> Signup and view all the answers

    What is the relationship between the vector $y$ and the orthogonal basis $E$?

    <p>Vector $y$ is a linear combination of the basis vectors in $E$.</p> Signup and view all the answers

    What does the term 'orthogonal projection' refer to in the context of vector spaces?

    <p>The process of finding the closest point in a subspace to a given vector.</p> Signup and view all the answers

    Which of the following best describes the span of a set of vectors?

    <p>The set of all possible linear combinations of the vectors.</p> Signup and view all the answers

    If $W$ is defined as Span($ ext{u}_1, ext{u}_2, ext{u}_3$), which of the following is true?

    <p>Every vector in $W$ can be expressed as a combination of $ ext{u}_1$, $ ext{u}_2$, and $ ext{u}_3$.</p> Signup and view all the answers

    What property must the basis of $W$ have to be considered orthogonal?

    <p>All pairs of vectors must be orthogonal to each other.</p> Signup and view all the answers

    What characterizes an orthogonal set of vectors?

    <p>The dot product of each pair of distinct vectors is zero.</p> Signup and view all the answers

    Given the vectors $u_1 = (1, 0)$ and $u_2 = (0, 1)$, are they orthogonal?

    <p>Yes, their dot product is 0.</p> Signup and view all the answers

    If vectors $u_i$ and $u_j$ are part of an orthogonal set, what can be inferred about their relationship?

    <p>Their dot product equals zero.</p> Signup and view all the answers

    Which of the following sets of vectors is orthogonal?

    <p>$u_1 = (2, 0)$, $u_2 = (0, 2)$</p> Signup and view all the answers

    What is the result of the dot product for two orthogonal vectors?

    <p>It is always 0.</p> Signup and view all the answers

    Which property does NOT belong to orthogonal sets of vectors?

    <p>They can only exist in two-dimensional space.</p> Signup and view all the answers

    Which statement is true regarding orthogonal sets of vectors?

    <p>The vectors can have any length.</p> Signup and view all the answers

    In the context of orthogonal sets, the notation $U_i . U_j = 0$ signifies what?

    <p>The vectors are orthogonal.</p> Signup and view all the answers

    What is the value of 'u' when calculated from the equation provided in the content?

    <p>40</p> Signup and view all the answers

    What is the result of the expression $2(2)$ based on the content?

    <p>4</p> Signup and view all the answers

    If $5(i) = y - j$, what is 'j' given that 'y' is expressed as $5i$?

    <p>$j = y - 5i$</p> Signup and view all the answers

    What geometric concept is related to the 'Span' in the context provided?

    <p>Linear combinations</p> Signup and view all the answers

    For values expressed in the format $||y - y_a|| + 2 = 5$, what does 'y_a' represent?

    <p>An arbitrary point in the space</p> Signup and view all the answers

    What does the notation $||y - y||$ equal to?

    <p>0</p> Signup and view all the answers

    Which of the following expressions represents a line orthogonal to the vector 'u'?

    <p>$(-7, 5)$</p> Signup and view all the answers

    Given $nu = 2$ as stated in the content, what is the possible conclusion for 'nu'?

    <p>Nu must be double the value of 'u'</p> Signup and view all the answers

    If 'y' is expressed in terms of 'z' as $y = 4z$, what does 'y' depend on?

    <p>It varies directly with 'z'</p> Signup and view all the answers

    What does the expression $||y - L|| = x$ signify in geometric terms?

    <p>Distance from point 'y' to line L</p> Signup and view all the answers

    What condition must a matrix U meet to have orthonormal columns?

    <p>UTU = I_n</p> Signup and view all the answers

    If U is a mxn matrix with orthonormal columns, what is true about the inner product of two vectors x and y?

    <p>(Ux)(Uy) = 0 iff x . y = 0</p> Signup and view all the answers

    In the context of an mxn matrix U with orthonormal columns, which relationship correctly shows the transformation of vector x?

    <p>||Ux|| = ||x||</p> Signup and view all the answers

    What is a characteristic of an orthogonal square matrix U?

    <p>U^T = U^-1</p> Signup and view all the answers

    What is the outcome of Ux if U has orthonormal columns and x is a vector?

    <p>Ux preserves the length of x.</p> Signup and view all the answers

    Which statement about the inner product (Ux)(Uy) is true when U has orthonormal columns?

    <p>(Ux)(Uy) equals (x . y).</p> Signup and view all the answers

    If U is an mxn matrix with orthonormal columns, what does it imply about the columns of U?

    <p>Columns are linearly independent.</p> Signup and view all the answers

    What happens to the vector x when multiplied by an orthonormal matrix U?

    <p>The direction of x remains unchanged.</p> Signup and view all the answers

    Study Notes

    Orthogonal Sets

    • A set of vectors {u₁, u₂, ..., up} in ℝn is called an orthogonal set if each pair of distinct vectors from the set is orthogonal. This means ui ⋅ uj = 0 whenever i ≠ j.

    Example

    • Show that {u₁, u₂, u₃} is an orthogonal set, where u₁ = [3, -1, 1], u₂ = [1, 2, 2], and u₃ = [-1, -2, 7/2].

    • u₁ ⋅ u₂ = 3(-1) + 1(2) + 1(2) = -3 + 2 + 2 = 1 ≠ 0, which means the set is not orthogonal.

    Theorem

    • If S = {u₁, u₂, ..., up} is an orthogonal set of nonzero vectors in ℝn, then S is linearly independent. This means the vectors are not scalar multiples of each other.

    • Hence, S is a basis for the subspace spanned by S.

    Definition: Orthogonal Basis

    • An orthogonal basis for a subspace W of ℝn is a basis for W that is also an orthogonal set.

    Theorem (weights in a linear combination)

    • Let {u₁, u₂, ..., up} be an orthogonal basis for a subspace W of ℝn. For each y ∈ W, the weights in the linear combination y = c₁u₁ + c₂u₂ + ... + cpup are given by cj = (y ⋅ uj) / (uj ⋅ uj) for j = 1, ..., p.

    Example

    • The set {u₁, u₂, u₃}, where u₁ = [3, -1, -1/2], u₂ = [1, 2, 1], and u₃ = [1, -2, 7/2], is an orthogonal basis for ℝ³.

    • Express the vector y = [6, 1, -8] as a linear combination of the vectors in S.

    Example Solution

    • Calculate the dot products: y ⋅ u₁ = 11 y ⋅ u₂ = -12 y ⋅ u₃ = -33 u₁ ⋅ u₁ = 1 , u₂ ⋅ u₂ = 6 , u₃ ⋅ u₃ = 33/2
    • Thus, c₁ =(y ⋅ u₁)/ (u₁ ⋅ u₁) = 11/11 =1, and c₂ =(-12/6) = −2 and c₃ =(-33 / (33/2)) = -2.
    • y = 11u₁ - 12u₂ - 33u₃/11 (The actual calculations/ formula are part of this solution.
    • Note: There were issues with the provided example vectors. The set of vectors given for u₁, u₂ and u₃ are not mutually orthogonal.

    Orthogonal Projections

    • Given a vector y ∈ ℝn and a vector u ∈ ℝn, the orthogonal projection of y onto u is given by ŷ = (y⋅u)/(u⋅u) ⋅ u.

    • ŷ is a vector in the subspace spanned by u.

    • The component of y orthogonal to u is y - ŷ.

    Theorem

    • An m × n matrix U has orthonormal columns if and only if UTU = In

    • ||Ux|| = ||x|| if UTU = In

    • (Ux)⋅(Uy) = x⋅y if U has orthogonal columns

    • (Ux) ⋅ (Uy) = 0 if and only if x ⋅ y = 0

    Gram-Schmidt Process

    • An algorithm for producing an orthogonal/normal basis for any subspace of ℝn.

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    Description

    This quiz delves into the concept of orthogonal sets within the realm of linear algebra. It explores the definition, implications, and theorems related to orthogonality and linear independence of vectors. Test your understanding of these crucial concepts and their applications in subspaces of ℝⁿ.

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