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Linear Algebra(Aljabar Linier)
Week 10
Universitas Multimedia NusantaraSerpong, Tangerang
Dr. Ananda Kusumae-mail: [email protected]
Agenda
• Orthogonality– Orthogonality in Rn , Orthogonal complements, Orthogonal
Projections
– The Gram-Schmidt Process
– The QR Factorization• Approximating eigenvalues
– Orthogonal Diagonalization of Symmetric Matrices
• Vector Spaces– Vector spaces and subspaces
– Linear Independence, basis, and dimension
– Change of basis
– Linear Transformation: Kernel and Range
– The Matrix of a linear transformation
The Gram-Schmidt Processand
The QR Factorization
Constructing Orthogonal Vectors
•
The Gram-Schmidt Process
QR Factorization
• The Gram-Schmidt process has shown that for each i=1,...,n,
Example
• QR Factorization procedure:• Use the Gram-Schmidt process to find an orthonormal basis for Col A• Since Q has orthonormal columns, then . If then
• Find a QR factorization of
111
011
001
A
IQQ T QRA AQR T
Approximating Eigenvalues
• The idea is based on the following:
where
• All the Ak are similar and hence they have the same eigenvalues. Under certain conditions, the matrices Ak converge to a triangular matrix (the Schur form of A), where the eigenvalues are listed on the diagonal
• Example: Compute eigenvalues of
12
32A
kkkkkTkkkk
Tkkkk QAQQAQQRQQQRA 1
1
AA :0
Orthogonal Diagonalizationof Symmetric Matrices
Spectral Theorem
The spectral decomposition of AThe projection form of the Spectral Theorem
Example
• Find the spectral decomposition of the matrix
211
121
112
A
Vector Spaces&
Subspaces
• Definition: Let V be a set on which addition and scalar multiplication are defined. If the following axioms are true for all objects u, v, and w in V and all scalars c and d then V is called a vector space and the objects in V are called “vectors”
• Note: objects called vectors here are not only Euclidean vectors (previous lectures), but they can be matrices, functions, etc.
Vector Spaces
• Let the set V be the points on a line through the origin, with the standard addition and scalar multiplication. Show that V is a vector space .
• Let the set V be the points on a line that does NOT go through the origin in with the standard addition and scalar multiplication. Show that V is not a vector space
• Let n and m be fixed numbers and let represent the set of nxm all matrices. Also let addition and scalar multiplication on be the standard matrix addition and standard matrix scalar multiplication. Show that is a vector space
• Show that is a vector space:
Examples
nmMnmM
nmM
•
Operations on real-valued functions
• Theorem:
• Note: Every vector space, V, has at least two subspaces. Namely, V itself and (the zero space)
Subspaces
• Let W be the set of diagonal matrices of size nxn. Is this a subspace of Mnn ?
• Let be the set of all polynomials of degree n or less. Is this a subspace of , where is a set of real-valued functions on the interval ?
Examples
nP],[ baF ],[ baF
],[ ba
Examples:
Spanning Sets
Linear IndependenceBasis
Dimension
Examples:
Linear Independence
Examples:
Basis
Remark: The most useful aspect pf coordinate vectors is that they allow us to transfer information from a general vector space to Rn
Examples:
Coordinates
Examples:
Dimension
Change of Basis
Introduction
The End
Thank you for your attention!