MATH 232 Linear Algebra Spring 2005 Proof by induction Proof by
... In some cases, the list of statements S(n) is to be proven for all natural numbers n ≥ K for a fixed K. In these cases, one proves that S(K) is true. • Prove the implication: If S(k) is true, then S(k + 1) is true. The hypothesis “S(k) is true” is sometimes referred to as the inductive hypothesis. N ...
... In some cases, the list of statements S(n) is to be proven for all natural numbers n ≥ K for a fixed K. In these cases, one proves that S(K) is true. • Prove the implication: If S(k) is true, then S(k + 1) is true. The hypothesis “S(k) is true” is sometimes referred to as the inductive hypothesis. N ...
Math 54. Selected Solutions for Week 2 Section 1.4
... line in R2 through (4, 1) and the origin. Then, find a vector ~b in R2 such that the solution set of A~x = ~b is not a line in R2 parallel to the solution set of A~x = ~0 . Why does this not contradict Theorem 6? We can find a homogeneous linear equation in (x1 , x2 ) that has solution x1 = 4 , x2 = ...
... line in R2 through (4, 1) and the origin. Then, find a vector ~b in R2 such that the solution set of A~x = ~b is not a line in R2 parallel to the solution set of A~x = ~0 . Why does this not contradict Theorem 6? We can find a homogeneous linear equation in (x1 , x2 ) that has solution x1 = 4 , x2 = ...
Notes 11: Dimension, Rank Nullity theorem
... ones in the RREF of M. We examine our algorithm for finding a basis of im(M ). We start with the set of m column vectors of M and we remove some of them. Indeed we remove the columns corresponding to the free variables. These are the columns that do not have a leading one. The columns remaining are ...
... ones in the RREF of M. We examine our algorithm for finding a basis of im(M ). We start with the set of m column vectors of M and we remove some of them. Indeed we remove the columns corresponding to the free variables. These are the columns that do not have a leading one. The columns remaining are ...
matrix-vector multiplication
... we only store its lower triangular part. A straightforward approach in storing the elements of a skyline matrix is to place all the rows (in order) into a oating-point array (val(:)), and then keep an integer array (row ptr(:)) whose elements point to the beginning of each row. The column indices o ...
... we only store its lower triangular part. A straightforward approach in storing the elements of a skyline matrix is to place all the rows (in order) into a oating-point array (val(:)), and then keep an integer array (row ptr(:)) whose elements point to the beginning of each row. The column indices o ...
2.2 The Inverse of a Matrix
... Similarly, one can show that B −1 A −1 AB = I. Theorem 6, part b can be generalized to three or more invertible matrices: ABC −1 = __________ Earlier, we saw a formula for finding the inverse of a 2 × 2 invertible matrix. How do we find the inverse of an invertible n × n matrix? To answer th ...
... Similarly, one can show that B −1 A −1 AB = I. Theorem 6, part b can be generalized to three or more invertible matrices: ABC −1 = __________ Earlier, we saw a formula for finding the inverse of a 2 × 2 invertible matrix. How do we find the inverse of an invertible n × n matrix? To answer th ...
2.2 The Inverse of a Matrix The inverse of a real number a is
... Similarly, one can show that B −1 A −1 AB = I. Theorem 6, part b can be generalized to three or more invertible matrices: ABC −1 = __________ Earlier, we saw a formula for finding the inverse of a 2 × 2 invertible matrix. How do we find the inverse of an invertible n × n matrix? To answer th ...
... Similarly, one can show that B −1 A −1 AB = I. Theorem 6, part b can be generalized to three or more invertible matrices: ABC −1 = __________ Earlier, we saw a formula for finding the inverse of a 2 × 2 invertible matrix. How do we find the inverse of an invertible n × n matrix? To answer th ...
Calculus II - Basic Matrix Operations
... Adding two matrices is also done entry-by-entry. If A = (aij ) and B = (bij ) are two m × n matrices, then their sum is A + B = (aij + bij ). That is, the i, j-entry of A + B is the sum of the i, j-entries of A and B. It is important to note that is is only possible to add two matrices if they have ...
... Adding two matrices is also done entry-by-entry. If A = (aij ) and B = (bij ) are two m × n matrices, then their sum is A + B = (aij + bij ). That is, the i, j-entry of A + B is the sum of the i, j-entries of A and B. It is important to note that is is only possible to add two matrices if they have ...
Lecture 16 - Math TAMU
... to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they have the same (i) determinant, (ii) trace = the su ...
... to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they have the same (i) determinant, (ii) trace = the su ...
A Tutorial on MATLAB Objective: To generate arrays in MATLAB
... 3. Objective: To decide magnitude of the numbers and compare the numbers according to their magnitudes in MATLAB. Procedure: In MATLAB, there are six relational operations to make comparison between arrays. These operators are shown in Table 1. The result of a comparison using the relational operato ...
... 3. Objective: To decide magnitude of the numbers and compare the numbers according to their magnitudes in MATLAB. Procedure: In MATLAB, there are six relational operations to make comparison between arrays. These operators are shown in Table 1. The result of a comparison using the relational operato ...
Sept. 3, 2013 Math 3312 sec 003 Fall 2013
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
Matrix Inverses Suppose A is an m×n matrix. We have learned that
... As Example 5, p. 122, indicates, performing an elementary row operation on an n × n matrix A can be represented in terms of matrix multiplication: if we perform a certain row operation to the n × n identity I n and obtain the € matrix E (called an elementary matrix), then performing the same row op ...
... As Example 5, p. 122, indicates, performing an elementary row operation on an n × n matrix A can be represented in terms of matrix multiplication: if we perform a certain row operation to the n × n identity I n and obtain the € matrix E (called an elementary matrix), then performing the same row op ...
MATHEMATICAL METHODS SOLUTION OF LINEAR SYSTEMS I
... – interchange of the i-th and j-th rows – interchange of the i-th and j-th columns – multiplication of the i-th row by the non-zero constant k – multiplication of the i-th column by the non-zero constant k – addition to the i-th row the product of k times the j-th row – addition to the i-th column t ...
... – interchange of the i-th and j-th rows – interchange of the i-th and j-th columns – multiplication of the i-th row by the non-zero constant k – multiplication of the i-th column by the non-zero constant k – addition to the i-th row the product of k times the j-th row – addition to the i-th column t ...
Lecture 35: Symmetric matrices
... networks as learning maps x 7→ sign(W x) or in graph theory as adjacency matrices. Symmetric matrices play the same role as the real numbers do among the complex numbers. Their eigenvalues often have physical or geometrical interpretations. One can also calculate with symmetric matrices like with nu ...
... networks as learning maps x 7→ sign(W x) or in graph theory as adjacency matrices. Symmetric matrices play the same role as the real numbers do among the complex numbers. Their eigenvalues often have physical or geometrical interpretations. One can also calculate with symmetric matrices like with nu ...
