SECTION 2.2 MATRIX INVERSES If the matrix A is square and there is a matrix C such that CA = AC = I, then A is called invertible and C is called the inverse of A, which we write as A-1 . A matrix that is not invertible is called singular. FACT. Products a
SECTION 3.2 PROPERTIES OF DETERMINANTS Adding a multiple of one row to another row does not change the determinant. Multiplying a row by a constant also multiplies the determinant by that constant. Switching two rows multiplies the determinant by -1. A sq
SECTION 4.4 COORDINATE SYSTEMS Today, the Santa Claus development begins to thin again. Suppose we have any vector space V with a basis B = cfw_b1 , . . . , bp . Pause a moment to ponder what this might mean. Now, suppose we have any vector x in V . The c
SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY SOME MOTIVATION. (1) Suppose we want to solve Ax = b, but the system is inconsistent. Then we'd like to do the best we can, that is, we'd like to find an x so that Ax is as close as possible to b. We need t
SECTION 1.2 ROW REDUCTION AND ECHELON FORMS Our aim is to transform a system of linear equations into an equivalent system from which the solutions can easily be read. It's easiest to do this by performing row operations on the augmented matrix of the sys
SECTION 1.3 VECTOR EQUATIONS Until further notice, a vector will be a matrix with exactly one column. Vectors with two rows can be identified with points in the plane; vectors with three rows can be identified with points in three-dimensional space. The c
SECTION 1.5 SOLUTION SETS OF LINEAR SYSTEMS Systems of linear equations can always be written as a matrix equation Ax = b. When b = 0 we say the system is homogeneous. Homogeneous systems always have at least one solution. The question is usually whether
SECTION 1.6 APPLICATIONS EXAMPLE. Use a vector equation approach to balance the following chemical reaction. P bN6 + CrM n2 O8 P b3 O4 + Cr2 O3 + M nO2 + N O
EXAMPLE. Find the general flow pattern in the network shown below. Then, assuming that the flow m
SECTION 1.7 LINEAR INDEPENDENCE We know that 3 vectors in R4 cannot span R4 . What about 4 vectors, say v1 , v2 , v3 , v4 ? If, say, v4 is a LC of the others, then what can we say about Spancfw_v1 , v2 , v3 , v4 ?
LINEARLY DEPENDENT LISTS OF VECTORS. A li
SECTION 1.8 ANOTHER WAY TO THINK ABOUT Ax = b
Ax = b is a way to write a system of equations. Here's the new way to think about Ax . Given a vector x, then Ax is another vector y. When the matrix A is m n, then we get a function or transformation T from
SECTION 2.1 MATRIX ALGEBRA You can multiply matrices by scalars (numbers); when two matrices are the same size you can add and subtract them; when two matrices are appropriate sizes you can multiply them. We know about the matrix-vector product Ax, so to
SECTION 3.1 DETERMINANTS A square matrix A has a determinant, denoted by det A. To calculate det A, we start with the 2 2 case: a b det = ad - bc c d For any matrix A, when we cross out row i and column j, we get a new matrix denoted by Aij . Then for any