Linear Algebra Review. Vectors
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1 Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors The length of x, a.k.a. the norm or 2-norm of x, is x = x 2 + x 2 2 +L+ x n 2 e.g., x = = 38 Tim Marks
2 Linear Algebra Review 9/4/7 Good Review Materials (Gonzales Woods review materials) Chapt. : Linear Algebra Review Chapt. 2: Probability, Random Variables, Random Vectors Online vector addition demo: Tim Marks 2
3 Linear Algebra Review 9/4/7 Vector Addition v u+v u Vector Subtraction u u-v v Tim Marks 3
4 Linear Algebra Review 9/4/7 Example (on board) Inner product (dot product) of two vectors 6 ( a = 2 ( 3 ( 4 b = 5 a b = a T b = [ 6 2 3] 4 ( ( 5 ( = (3) 5 = Tim Marks 4
5 Linear Algebra Review 9/4/7 Inner (dot) Product v α u The inner product is a SCALAR. u T v = u v Tim Marks 5
6 Linear Algebra Review 9/4/7 Transpose of a Matrix Transpose: C mn = A T nm c ij = a ji (A + B) T = A T + B T (AB) T = B T A T Examples: T = T = If A T = A, we say A is symmetric. Example of symmetric matrix Tim Marks 6
7 Linear Algebra Review 9/4/7 Tim Marks 7
8 Linear Algebra Review 9/4/7 Tim Marks 8
9 Linear Algebra Review 9/4/7 Tim Marks 9
10 Linear Algebra Review 9/4/7 Product: C n p = A nm B m p Matrix Product A and B must have compatible dimensions c ij = m k= a ik b kj Examples: 2 5 ( = In Matlab: >> A*B 6 2 ( = Matrix Multiplication is not commutative: A nn B nn B nn A nn Tim Marks
11 Linear Algebra Review 9/4/7 Matrix Sum Sum: C nm = A nm + B nm c ij = a ij + b ij A and B must have the same dimensions Example: = Determinant of a Matrix Determinant: A must be square det a a 2 = a a 2 = a a 2 a 22 a 2 a a 22 ( a 2 a 2 22 Example: det 2 5 = 2 (5 = (3 3 a a 2 a 3 a 22 a 23 a 2 a 23 a 2 a 22 det a 2 a 22 a 23 = a ( a 2 + a 3 a a 3 a 32 a a 33 a 3 a 33 a 3 a 32 Tim Marks
12 Linear Algebra Review 9/4/7 Determinant in Matlab Inverse of a Matrix If A is a square matrix, the inverse of A, called A -, satisfies AA - = I and A - A = I, Where I, the identity matrix, is a diagonal matrix with all s on the diagonal. I 2 = I 3 = Tim Marks 2
13 Linear Algebra Review 9/4/7 Inverse of a 2D Matrix Example: ( = 5 (2 28 ( ( ) 6 2 = 5 (2 ) 6 2 = 28 = 5 28 ( Inverses in Matlab Tim Marks 3
14 Linear Algebra Review 9/4/7 Trace of a matrix Example (on board) Matrix Transformation: Scale A square, diagonal matrix scales each dimension by the corresponding diagonal element. Example: = Tim Marks 4
15 Linear Algebra Review 9/4/7 Tim Marks 5 Linear Independence A set of vectors is linearly dependent if one of the vectors can be expressed as a linear combination of the other vectors. Example:,, 2 A set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the other vectors. Example:,, 2 3 Rank of a matrix The rank of a matrix is the number of linearly independent columns of the matrix. Examples: has rank 2 2 Note: the rank of a matrix is also the number of linearly independent rows of the matrix. has rank 3 2
16 Linear Algebra Review 9/4/7 Tim Marks 6 Singular Matrix All of the following conditions are equivalent. We say a square (n n) matrix is singular if any one of these conditions (and hence all of them) is satisfied. The columns are linearly dependent The rows are linearly dependent The determinant = The matrix is not invertible The matrix is not full rank (i.e., rank < n) Linear Spaces A linear space is the set of all vectors that can be expressed as a linear combination of a set of basis vectors. We say this space is the span of the basis vectors. Example: R 3, 3-dimensional Euclidean space, is spanned by each of the following two bases:,,, 2,
17 Linear Algebra Review 9/4/7 Linear Subspaces A linear subspace is the space spanned by a subset of the vectors in a linear space. The space spanned by the following vectors is a two-dimensional subspace of R 3., What does it look like? The space spanned by the following vectors is a two-dimensional subspace of R 3., What does it look like? Orthogonal and Orthonormal Bases n linearly independent real vectors span R n, n-dimensional Euclidean space They form a basis for the space. An orthogonal basis, a,, a n satisfies a i a j = if i j An orthonormal basis, a,, a n satisfies a i a j = if i j a i a j = if i = j Examples. Tim Marks 7
18 Linear Algebra Review 9/4/7 Orthonormal Matrices A square matrix is orthonormal (also called unitary) if its columns are orthonormal vectors. A matrix A is orthonormal iff AA T = I. If A is orthonormal, A - = A T AA T = A T A = I. A rotation matrix is an orthonormal matrix with determinant =. It is also possible for an orthonormal matrix to have determinant =. This is a rotation plus a flip (reflection). 2D rotation matrix: Matrix Transformation: Rotation by an Angle θ Tim Marks 8
19 Linear Algebra Review 9/4/7 Tim Marks 9
20 Linear Algebra Review 9/4/7 Some Properties of Eigenvalues and Eigenvectors If λ,, λ n are distinct eigenvalues of a matrix, then the corresponding eigenvectors e,, e n are linearly independent. If e is an eigenvector of a matrix with corresponding eigenvalue λ, then any nonzero scalar multiple of e is also an eigenvector with eigenvalue λ. A real, symmetric square matrix has real eigenvalues, with orthogonal eigenvectors (can be chosen to be orthonormal). SVD: Singular Value Decomposition Any matrix A (m n) can be written as the product of three matrices: A = UDV T where U is an m m orthonormal matrix D is an m n diagonal matrix. Its diagonal elements, σ, σ 2,, are called the singular values of A, and satisfy σ σ 2. V is an n n orthonormal matrix Example: if m > n A U D V T ( ( ( ( * * 2 T + v, = u u 2 u 3 L u m * n M M M T + v n, ) ) ) ) Tim Marks 2
21 Linear Algebra Review 9/4/7 >> a = [ 2 3; 2 7 4; -3 6; 2 4 9; 5-8 ] a = SVD in Matlab >> [u,d,v] = svd(a) u = d = v = Some Properties of SVD The rank of matrix A is equal to the number of nonzero singular values σ i A square (n n) matrix A is singular if and only if at least one of its singular values σ,, σ n is zero. Tim Marks 2
22 Linear Algebra Review 9/4/7 Geometric Interpretation of SVD If A is a square (n n) matrix, A U D V T = ( L ( u L u n ) L ) * T + v, * 2 M M M T * n + v n, U is a unitary matrix: rotation (possibly plus flip) D is a scale matrix V (and thus V T ) is a unitary matrix Punchline: An arbitrary n-d linear transformation is equivalent to a rotation (plus perhaps a flip), followed by a scale transformation, followed by a rotation Advanced: y = Ax = UDV T x V T expresses x in terms of the basis V. D rescales each coordinate (each dimension) The new coordinates are the coordinates of y in terms of the basis U Tim Marks 22
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