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Introduction
• Xin Liu• PhD student of Dr. Rokne
• Contact• [email protected]
• Slides downloadable at• pages.cpsc.ucalgary.ca/~liuxin
• The way to math world• Lecture attendance
• Hard to learn by yourselves
• Practices, practices, and practices …
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Matrix-Vector Multiplication
• Linear (the 1st degree) systems are the simplest, but most widely used systems in science and engineering
• A basic problem: solving the linear equation system
• Straight forward method• Gaussian elimination
• Hard to do because• large scale• poor conditioned
• small disturbance in coefficients causes big difference in solutions
• A better method• SVD – Singular Vale Decomposition• Will be introduced gradually in a series of lectures
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Definitions
• An n-vector is defined as
• Think about 3-vectors in Euclidean space
• An mxn matrix is defined as
• Multiplication
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Linear Mapping
• is a linear mapping, which satisfies• Distributive law• Associative law (for scalar)
• Conversely, every linear map from Rn to Rm can be expressed as a multiplication by an mxn matrix
•
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Mat-vect multiplication
• View the matrix-vector multiplication from another angle
• If we write A as a combination of column vectors
• Then the mat-vect multiplication can be written as
• That means: b is a linear combination of the columns of A
•
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Mat-mat multiplication
• Matrix-matrix multiplication
• is defined as
• We can calculate B columnwisely
• Each column of B is a linear combination of the columns aj with the coefficients ckj
•
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Range
• Definition:• The range of a matrix A, is the set of vectors
that can be expressed as Ax for some x.
• Theorem• range (A) is the space spanned by the columns
of A.
• The range of A is also called the column space of A.
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Nullspace
• Definition:• The nullspace (solution space) of A is
the set of vectors x that satisfy Ax = 0.
• Each vector x in the nullspace gives the expansion coefficients of the zero vector as a linear combination of columns of A
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Rank
• Column rank = dimension of space spanned by the matrix’s columns = # of linearly independent columns
• Row rank = dimension of space spanned by the matrix’s rows = # of linearly independent rows
• Row rank = Column rank = Matrix rank
• Full rank
• Theorem
•
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Inverse
• A nonsingular or invertible matrix must be square and full rank.
• The m columns of a nonsingular mxm matrix A span (form a basis) for the whole space Rm
• Any vector in Rm can be expressed as a linear combination of the columns of A
• The inverse of A is a matrix A-1, such that• AA-1 = A-1A = I• I is the mxm identity matrix
• The inverse of a nonsingular matrix is unique.
• A-1b is the unique solution of Ax = b.
• A-1b is the vector of coefficients of the expansion of b in the basis of the columns of A.
•
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Transpose
• Definition• The transpose AT of an mxn matrix A is nxm where the (i,j)
entry of AT is the (j, i) entry of A.
• Example
• A is symmetric if A = AT.
• Multiplication
• •
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Orthogonal vectors
• Orthogonal (perpendicular) vectors• Vectors x, y are orthogonal if xTy = 0.
• Orthogonal vector set
• Orthogonal two vector sets
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Components of a vector
• Inner products can be used to decompose arbitrary vectors into orthogonal components (project onto orthonormal vectors).
•