Linear-Algebra - Operations On One Matrix

Operations on one matrix Solving linear systems Solving systems with substitution 1. Get a variable by itself in one of the equations. 2. Take the expression you got for the variable in step 1, and plug it (substitute it using parentheses) into the other equation. 3. Solve the equation in step 2 for the remaining variable. 4. Use the result from step 3 and plug it into the equation from step 1. Solving systems with elimination 1. If necessary, rearrange both equations so that the x-terms are first, followed by the y-terms, the equals sign, and the constant term (in that order). If an equation appears to have not constant term, that means that the constant term is 0. 2. Multiply one (or both) equations by a constant that will allow either the x-terms or the y-terms to cancel when the equations are added or subtracted (when their left sides and their right sides are added separately, or when their left sides and their right sides are subtracted separately). 3. Add or subtract the equations. 4. Solve for the remaining variable. 1 5. Plug the result of step 4 into one of the original equations and solve for the other variable. Graphing method 1. Solve for y in each equation. 2. Graph both equations on the same Cartesian coordinate system. 3. Find the point of intersection of the lines (the point where the lines cross). Matrix dimensions and entries “rows × columns” K= [k2,1 k2,2 k2,3] k1,1 k1,2 k1,3 Augmented matrix M= [m2,1 m2,2 m2,3 m2,4 | C2] m1,1 m1,2 m1,3 m1,4 | C1 Pivot entries and pivot columns Pivot entry, leading entry, pivot: the first non-zero entry in each row. 2 Pivot column: Any column that houses a pivot. Echelon forms Row-echelon form (ref) 1. All the pivot entries are equal to 1. 2. Any row(s) that consist of only 0s are at the bottom of the matrix. 3. The pivot in each row sits in a column to the right of the column that houses the pivot in the row above it. In other words, the pivot entries sit in a staircase pattern, where they stair-step down from the upper left corner to the lower right corner of the matrix. Reduced row-echelon form (rref) 1. All the pivot entries are equal to 1. 2. Any row(s) that consist of only 0s are at the bottom of the matrix. 3. The pivot in each row sits in a column to the right of the column that houses the pivot in the row above it. In other words, the pivot entries sit in a staircase pattern, where they stair-step down from the upper left corner to the lower right corner of the matrix. 4. Every pivot is the only non-zero entry in its column. Gauss-Jordan elimination, Gaussian elimination 3 1. Optional: Pull out any scalars from each row in the matrix. 2. If the first entry in the first row is 0, swap it with another row that has a non-zero entry in its first column. Otherwise, move to step 3. 3. Multiply through the first row by a scalar to make the leading entry equal to 1. 4. Add scaled multiples of the first row to every other row in the matrix until every entry in the first column, other than the leading 1 in the first row, is a 0. 5. Go back step 2 and repeat the process until the matrix is in reduced row-echelon form. Number of solutions to the system One unique solution: 1 0 0 | a 0 1 0 | b 0 0 1 | c No solutions: 1 0 0 | a 0 1 0 | b 0 0 0 | c Infinitely many solutions: 1 0 0 | a 0 1 0 | b 0 0 0 | 0 4 Operations on two matrices Matrix addition • Matrix dimensions must be identical • Add corresponding matrix entries to find the sum • Matrix addition is commutative and associative Matrix subtraction • Matrix dimensions must be identical • Subtract corresponding matrix entries to find the sum • Matrix subtraction is not commutative and not associative Scalar multiplication Scalar: a constant that gets multiplied by every entry in the matrix Opposite matrices A and − A 5 Matrix multiplication The number of columns in the first matrix must match the number of rows in the second matrix. In the product of A and B, with b1,1 b1,2 a1,1 a1,2 and B = A= a a [ 2,1 2,2] [ b2,1 b2,2] the entries of AB are (AB)1,1 is the product of the first row and first column (AB)2,1 is the product of the second row and first column (AB)1,2 is the product of the first row and second column (AB)2,2 is the product of the second row and second column The dimensions of the product of any A and B are the rows of the first matrix by the columns of the second matrix. Matrix multiplication is associative and distributive, but not commutative. Zero and identity matrices Zero matrix: A matrix with only 0 entries 6 Identity matrix: A matrix with only 1 entries 1 0 I2 = [ 0 1] 1 0 0 I3 = 0 1 0 [ 0 0 1] 1 0 I4 = 0 0 0 1 0 0 0 0 1 0 0 0 0 1 The product of the identity matrix and a matrix A: IA = A, but I must have the same number of columns as A has rows AI = A, but I must have the same number of rows as A has columns Matrices as vectors Vectors Vector: A vector has two pieces of information contained within it: 1. the direction in which the vector points, and 2. the magnitude of the vector, which is just the length of the vector. Row and column vectors Column vector: A vector expressed as a one-column matrix Row vector: A vector expressed as a one-row matrix 7 Sketching vectors Initial point: The point where the vector begins Terminal point: the point where the vector ends Standard position: A vector sketched in standard position has its initial point at the origin Vector operations  Sum of vectors: a  + b = (a1, a2) + (b1, b2) = (a1 + b1, a2 + b2)  Difference of vectors: a  − b = (a1, a2) − (b1, b2) = (a1 − b1, a2 − b2) Unit vectors and basis vectors Unit vector: a vector with length 1 u = 1 v || v  || || = Vector length: | | v  v12 + v22 + v32 + . . . + vn2 Standard basis vectors 8 For ℝ2: i = (1,0) and j = (0,1) For ℝ3: i = (1,0,0) and j = (0,1,0) and k = (0,0,1) Linear combinations and span Linear combination: The sum of scaled vectors Span of a vector set: The collection of all vectors which can be represented by linear combinations of the set. Linear dependence and independence Linear dependence: A set of vectors are linearly dependent when one vector in the set can be represented by a linear combination of the other vectors in the set. Collinear: Parallel vectors, or vectors that lie along the same line, are collinear. Coplanar: Parallel vectors, or vectors that lie in the same plane, are coplanar. Test for linear independence If (c1, c2, c3, . . . cn) = (0,0,0...0) is the only solution to  c1 v 1+ c2 v 2+ c3 v 3+ . . . + cn v n= O n, then the vector set V = {v1, v2, v3, . . . vn} is 9