Vectors and Vector Operations
... There are various types of objects to which the term vector is applied. These include row and column vectors, matrices, functions and directed line segments. We begin by looking at each of these and discuss the features they have in common which is why they are all called vectors. Row and column vec ...
... There are various types of objects to which the term vector is applied. These include row and column vectors, matrices, functions and directed line segments. We begin by looking at each of these and discuss the features they have in common which is why they are all called vectors. Row and column vec ...
lab chapter 5: simultaneous equations
... vectors. According to (5.30), the angle between two vectors is π/2 when xT y = 0. That is, vectors are orthogonal when their inner product (5.12 and 5.13) is zero. Norms can be used to determine how similar two vectors are. It was easy to see that equations (5.2) were singular because they are a sma ...
... vectors. According to (5.30), the angle between two vectors is π/2 when xT y = 0. That is, vectors are orthogonal when their inner product (5.12 and 5.13) is zero. Norms can be used to determine how similar two vectors are. It was easy to see that equations (5.2) were singular because they are a sma ...
Sketching as a Tool for Numerical Linear Algebra
... How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = AT(Ax-Ax’) = 0 Normal Equation: ATAx = ATb for any opti ...
... How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = AT(Ax-Ax’) = 0 Normal Equation: ATAx = ATb for any opti ...
Matrix Operations
... The empty matrix is represented in MATLAB as [ ]. This is a matrix with dimension zero-by-zero. Therefore, the statement >>B= [ ] enters a zero-by-zero matrix B into the workspace. The empty matrix has some special uses. It can serve, for example, to reduce the size of an existing matrix. For examp ...
... The empty matrix is represented in MATLAB as [ ]. This is a matrix with dimension zero-by-zero. Therefore, the statement >>B= [ ] enters a zero-by-zero matrix B into the workspace. The empty matrix has some special uses. It can serve, for example, to reduce the size of an existing matrix. For examp ...
3.4 Solving Matrix Equations with Inverses
... in the rows of the first matrix with the numbers in the second matrix as well as the sum. Instead of multiplying the first matrix by a constant column matrix, let’s try multiplying by a matrix containing two variables x and y, ...
... in the rows of the first matrix with the numbers in the second matrix as well as the sum. Instead of multiplying the first matrix by a constant column matrix, let’s try multiplying by a matrix containing two variables x and y, ...
Inner products and projection onto lines
... The nullspace of A is defined by solutions to A.x = 0. If we transform A.x = 0 to U.x = 0 by a set of row operations then it is obvious that solutions to U.x = 0 must be the same as those to A.x = 0 and so the nullspace of A is the same as the nullspace of U. It has dimension n – r (nullity), i.e. t ...
... The nullspace of A is defined by solutions to A.x = 0. If we transform A.x = 0 to U.x = 0 by a set of row operations then it is obvious that solutions to U.x = 0 must be the same as those to A.x = 0 and so the nullspace of A is the same as the nullspace of U. It has dimension n – r (nullity), i.e. t ...
Solutions to Homework 2
... Solution: Consider the coordinate mapping T : W → R7 defined by T (f ) = [f ]β , where [f ]β is the coordinate vector of f with respect to the basis β. Then α is linearly independent in W if and only if the set T (α) = {T (1), T (cos x), T (cos 2x), . . . , T (cos 6x)} is linearly independent in R7 ...
... Solution: Consider the coordinate mapping T : W → R7 defined by T (f ) = [f ]β , where [f ]β is the coordinate vector of f with respect to the basis β. Then α is linearly independent in W if and only if the set T (α) = {T (1), T (cos x), T (cos 2x), . . . , T (cos 6x)} is linearly independent in R7 ...
Linear Algebra, Section 1.9 First, some vocabulary: A function is a
... is never mentioned anymore). Normally, the question is whether the function is onto its codomain. For example, y = x2 is not onto the real line, but is onto its range, which is the interval [0, ∞). If we don’t want to specify that a function is onto its codomain, we will say that f maps x into the c ...
... is never mentioned anymore). Normally, the question is whether the function is onto its codomain. For example, y = x2 is not onto the real line, but is onto its range, which is the interval [0, ∞). If we don’t want to specify that a function is onto its codomain, we will say that f maps x into the c ...
Linear models 2
... A matrix is a rectangular array of numbers. We denote the vector space of m × n matrices by Rm×n . We denote the ith row of a matrix A by Ai: , the jth column by A:j and the (i, j) entry by Aij . The transpose of a matrix is obtained by switching its rows and columns. Definition 3.1 (Transpose). The ...
... A matrix is a rectangular array of numbers. We denote the vector space of m × n matrices by Rm×n . We denote the ith row of a matrix A by Ai: , the jth column by A:j and the (i, j) entry by Aij . The transpose of a matrix is obtained by switching its rows and columns. Definition 3.1 (Transpose). The ...
Theorems and counterexamples on structured
... its spectrum lies entirely in the open right (left) half plane. In the sequel, the term ‘positive stable’ will be usually shortened to simply ‘stable’. Hermitian positive definite and totally nonnegative matrices are obviously stable (having only positive eigenvalues), while the stability of M-matri ...
... its spectrum lies entirely in the open right (left) half plane. In the sequel, the term ‘positive stable’ will be usually shortened to simply ‘stable’. Hermitian positive definite and totally nonnegative matrices are obviously stable (having only positive eigenvalues), while the stability of M-matri ...
examples of Markov chains, irreducibility and
... Thinking of q, qn as row vectors, the last two equations can be written via matrix multiplication from the right, qn+1 = qn p, qn = q0 pn . Writing q, qn instead as column vectors, and using matrix multiplication by the transpose matrix, from the left, the equations become qn+1 = pt qn , qn = (pt )n ...
... Thinking of q, qn as row vectors, the last two equations can be written via matrix multiplication from the right, qn+1 = qn p, qn = q0 pn . Writing q, qn instead as column vectors, and using matrix multiplication by the transpose matrix, from the left, the equations become qn+1 = pt qn , qn = (pt )n ...