Review Sheet 1 I. Matrix Operations A. Matrix entries 1. Dimension 2. i,j entry B. Rows, Columns, Entries C. Addition, Multiplication, Scalar Multiplication 1. Laws of combination (associative, distributive, etc.) 2. Multiplication as a Dot Product 3. A[b1, … , bp] = [Ab1, … , Abp], where bi is the ith column of B 4. A[x1, … , xn]T = A1x1 + … + Anxn where Ai is the ith column of A D. Transpose 1. Rows and columns interchanged E. Special Matrices 1. Diagonal matrix 2. Identity Mtrix 3. Upper Triangular 4. Lower Triangular F. Partitioned Matrices 1. Block Diagonal 2. Multiplying partitioned matrices a. Works like multiplication with numerical entries b. Dimensions must match Vectors A. Geometric Vectors 1. Direction 2. Length 3. Diagram a. Addition b. Subtraction c. Multiplication by a scalar Elementary Matrices A. Determining Elementary matrices that will have a certain effect 1. Interchange Rows -Type I 2. Multiply a Row by a Nonzero Constant - Type II 3. Add Row i to Row j for i not equal to j - Type III 4. Reading the action of an elementary matrix on a column vector B. Upper Triangular Matrix 1. Echelon Form and Reduced Echelon Form 2. Use in Solving Linear Equations a. Reading off a solution of a system of equations using back substitution i. Infinitely many solutions ii. Indicator for no solution 3. Inversion of upper triangular matrix close to the identity C. Lower triangular matrix
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Writing down a lower triangular matrix that does a certain Gaussian operation Systems of Linear Equations A. Writing the system in terms of a matrix equation 1. Coefficient Matrix 2. Augmented Matrix B. Finding an equivalent system with the coefficient matrix in upper triangular form 1. Gaussian Elimination C. Reading of the solution of a system of equations whose coefficient matrix is in upper triangular form 1. Variables s, t, etc. in the solution set 2. One or more rows of zeroes in the coefficient matrix 3. Substitution D. Reduced homogeneous system 1. Solution space = Null space (or kernel) of the coefficient matrix a. Vector Space 2. Solution of general system = particular solution + all solutions of the reduced homogeneous system 3. Solution space using variables in back substitution a. Using 1’s and 0’s in the non pivot position 3. Dimension of solution space 1. no. of columns - rank
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