Podcast
Questions and Answers
What characterizes the set Y as an orthogonal basis for W?
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?
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?
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?
What is the outcome of constructing an orthogonal basis Ev r23 for W?
When referring to the expression v = (2), what does the number 2 represent?
When referring to the expression v = (2), what does the number 2 represent?
What condition must be met for a basis of W to be considered orthogonal?
What condition must be met for a basis of W to be considered orthogonal?
In the context of orthogonal bases, which expression properly represents the summation of vectors?
In the context of orthogonal bases, which expression properly represents the summation of vectors?
Which interpretation can be drawn from the statement involving E and the orthogonal basis?
Which interpretation can be drawn from the statement involving E and the orthogonal basis?
If a basis is orthogonal, what can be said about the angles between the basis vectors?
If a basis is orthogonal, what can be said about the angles between the basis vectors?
Which of the following is true for an orthogonal basis concerning vector components?
Which of the following is true for an orthogonal basis concerning vector components?
What is the relationship between the vector $y$ and the orthogonal basis $E$?
What is the relationship between the vector $y$ and the orthogonal basis $E$?
What does the term 'orthogonal projection' refer to in the context of vector spaces?
What does the term 'orthogonal projection' refer to in the context of vector spaces?
Which of the following best describes the span of a set of vectors?
Which of the following best describes the span of a set of vectors?
If $W$ is defined as Span($ ext{u}_1, ext{u}_2, ext{u}_3$), which of the following is true?
If $W$ is defined as Span($ ext{u}_1, ext{u}_2, ext{u}_3$), which of the following is true?
What property must the basis of $W$ have to be considered orthogonal?
What property must the basis of $W$ have to be considered orthogonal?
What characterizes an orthogonal set of vectors?
What characterizes an orthogonal set of vectors?
Given the vectors $u_1 = (1, 0)$ and $u_2 = (0, 1)$, are they orthogonal?
Given the vectors $u_1 = (1, 0)$ and $u_2 = (0, 1)$, are they orthogonal?
If vectors $u_i$ and $u_j$ are part of an orthogonal set, what can be inferred about their relationship?
If vectors $u_i$ and $u_j$ are part of an orthogonal set, what can be inferred about their relationship?
Which of the following sets of vectors is orthogonal?
Which of the following sets of vectors is orthogonal?
What is the result of the dot product for two orthogonal vectors?
What is the result of the dot product for two orthogonal vectors?
Which property does NOT belong to orthogonal sets of vectors?
Which property does NOT belong to orthogonal sets of vectors?
Which statement is true regarding orthogonal sets of vectors?
Which statement is true regarding orthogonal sets of vectors?
In the context of orthogonal sets, the notation $U_i . U_j = 0$ signifies what?
In the context of orthogonal sets, the notation $U_i . U_j = 0$ signifies what?
What is the value of 'u' when calculated from the equation provided in the content?
What is the value of 'u' when calculated from the equation provided in the content?
What is the result of the expression $2(2)$ based on the content?
What is the result of the expression $2(2)$ based on the content?
If $5(i) = y - j$, what is 'j' given that 'y' is expressed as $5i$?
If $5(i) = y - j$, what is 'j' given that 'y' is expressed as $5i$?
What geometric concept is related to the 'Span' in the context provided?
What geometric concept is related to the 'Span' in the context provided?
For values expressed in the format $||y - y_a|| + 2 = 5$, what does 'y_a' represent?
For values expressed in the format $||y - y_a|| + 2 = 5$, what does 'y_a' represent?
What does the notation $||y - y||$ equal to?
What does the notation $||y - y||$ equal to?
Which of the following expressions represents a line orthogonal to the vector 'u'?
Which of the following expressions represents a line orthogonal to the vector 'u'?
Given $nu = 2$ as stated in the content, what is the possible conclusion for 'nu'?
Given $nu = 2$ as stated in the content, what is the possible conclusion for 'nu'?
If 'y' is expressed in terms of 'z' as $y = 4z$, what does 'y' depend on?
If 'y' is expressed in terms of 'z' as $y = 4z$, what does 'y' depend on?
What does the expression $||y - L|| = x$ signify in geometric terms?
What does the expression $||y - L|| = x$ signify in geometric terms?
What condition must a matrix U meet to have orthonormal columns?
What condition must a matrix U meet to have orthonormal columns?
If U is a mxn matrix with orthonormal columns, what is true about the inner product of two vectors x and y?
If U is a mxn matrix with orthonormal columns, what is true about the inner product of two vectors x and y?
In the context of an mxn matrix U with orthonormal columns, which relationship correctly shows the transformation of vector x?
In the context of an mxn matrix U with orthonormal columns, which relationship correctly shows the transformation of vector x?
What is a characteristic of an orthogonal square matrix U?
What is a characteristic of an orthogonal square matrix U?
What is the outcome of Ux if U has orthonormal columns and x is a vector?
What is the outcome of Ux if U has orthonormal columns and x is a vector?
Which statement about the inner product (Ux)(Uy) is true when U has orthonormal columns?
Which statement about the inner product (Ux)(Uy) is true when U has orthonormal columns?
If U is an mxn matrix with orthonormal columns, what does it imply about the columns of U?
If U is an mxn matrix with orthonormal columns, what does it imply about the columns of U?
What happens to the vector x when multiplied by an orthonormal matrix U?
What happens to the vector x when multiplied by an orthonormal matrix U?
Flashcards
Orthogonal Set
Orthogonal Set
A set of vectors in R^n is called an orthogonal set if every pair of distinct vectors in the set is orthogonal.
Orthogonal Vectors
Orthogonal Vectors
Two vectors are orthogonal if their dot product is zero.
Dot Product
Dot Product
The dot product of two vectors in R^n is calculated by multiplying corresponding components and summing the results.
Dot Product Formula
Dot Product Formula
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Vector in R^n
Vector in R^n
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Dot Product of a Vector with Itself
Dot Product of a Vector with Itself
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Linearly Independent Vectors
Linearly Independent Vectors
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Proof of Orthogonality
Proof of Orthogonality
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Orthogonal basis
Orthogonal basis
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Vector in the subspace
Vector in the subspace
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Projection onto a subspace
Projection onto a subspace
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Orthogonal component
Orthogonal component
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Pythagorean theorem in vector spaces
Pythagorean theorem in vector spaces
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Orthonormal Column Invertible Matrix
Orthonormal Column Invertible Matrix
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Norm Preservation in Orthonormal Matrix Transformations
Norm Preservation in Orthonormal Matrix Transformations
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Inner Product Preservation in Orthonormal Matrix Transformations
Inner Product Preservation in Orthonormal Matrix Transformations
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Orthogonal Matrix
Orthogonal Matrix
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Invertibility of Orthogonal Matrices
Invertibility of Orthogonal Matrices
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Orthonormal Column Condition for Orthogonal Matrices
Orthonormal Column Condition for Orthogonal Matrices
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Orthogonal projection
Orthogonal projection
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Span
Span
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Component of a vector
Component of a vector
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Decomposition of a vector
Decomposition of a vector
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Gram-Schmidt Process
Gram-Schmidt Process
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Linear Combination
Linear Combination
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Spanning Set
Spanning Set
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Normal Vector
Normal Vector
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Distance from Point to Line
Distance from Point to Line
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Projection of a Vector
Projection of a Vector
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Vector Length
Vector Length
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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.
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ŷ is a vector in the subspace spanned by u.
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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
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||Ux|| = ||x|| if UTU = In
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(Ux)⋅(Uy) = x⋅y if U has orthogonal columns
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(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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