Chapter 1 Flashcards
augmented matrix
the matrix that contains all of the numerical information contained in the system, including constants unassociated with variables
coefficient matrix
the matrix which contains the coefficients of the variables in a linear system
square matrix
if the number of rows of a matrix A equals the number of columns
diagonal matrix
a square matrix organized such that all entries above and below the main diagonal are zero
upper triangular matrix
square matrix organized sucht that all its entries below the main diagonal are zero
lower triangular matrix
square matrix organized sucht that all its entries above the main diagonal are zero
zero matrix
matrix with all entries = 0
vector
matrix with only one row/column; usually indicates a column vector
row/column vector
vector that is a single row/column
vector space, and how to denote
the set of all column vectors with n components, denoted with big R^n
standard representation of vectors
representing a vector within the Cartesian coordinate plane as an arrow (directed line segment) from the origin to the point (x, y). Default representation.
alternatives to standard representation
Can translate/shift the vector in the plane and connect some point (a, b) to the point (a + x, b + y).
gauss-jordan method
We proceed from equation to equation, from top to bottom.
Suppose we get to the ith equation. Let Xj be the leading variable of the
system consisting of the ith and all the subsequent equations.
• If Xj does not appear in the ith equation, swap the ith equation with the first equation below that does contain Xj.
• Suppose the coefficient of Xj in the ith equation is c; thus this equation is of
the form cxj + … = … Divide the ith equation by c.
• Eliminate xj from all the other equations, above and below the ith, by subtracting suitable multiples of the ith equation from the others.
Now proceed to the next equation. If an equation zero = nonzero emerges in this process, then the system fails to have solutions; the system is inconsistent.
When you are through without encountering an inconsistency, solve each
equation for its leading variable. You may choose the nonleading variables freely; the leading variables are then determined by these choices.
inconsistent system
a system of equations with no solutions
consistent system
a system of equations that has at least one solution
reduced row-echelon form
A matrix is in reduced row-echelon form if it satisfies all of the following conditions:
a. If a row has nonzero entries, then the first nonzero entry is a 1, called the leading 1 (or pivot) in this row.
b. If a column contains a leading 1, then all the other entries in that column are 0.
c. If a row contains a leading 1, then each row above it contains a leading 1 further to the left.
Condition c implies that rows of 0’s, if any, appear at the bottom of the matrix.
types of elementary row operations
- divide a row by a nonzero scalar
- subtract a multiple of a row from another row
- swap two rows
rank
the number of leading 1s in rref(A), denoted rank(A)
matrix
a rectangular array of numbers
possible number of solutions in a linear system
infinitely many solutions
exactly one solution
no solutions
number of equations (n) vs number of unknowns (m) and their impact on solutions
If a linear system as exactly one solution, then there must be at least as many equations as there are variables/unknowns. (m < m has either no solutions or infinitely many solutions.
when systems of n equations in n variables have a unique solution
only iff the rank of the coefficient matrix A is n
dot product of vectors v and w, and when possible
v1w1 + … vnwn. Must have same number of entries.
product of matrix M and vector V, and when possible
If M is n x m, then vector must be of length m (have as many rows as M has columns). The result will be a vector of height/rowcount n and width/columncount 1.
Two options:
- entry i of result will be the dot product of M’s row i with V
- result will be the sum of each product of V’s single-value entries with M’s column vectors.
It’s the linear combination of the columns of M with the components of V as the coefficients, if you get that reference. Which you should.
linear combination
a vector V in R^n is a linear combination of the vectors v1..vm in R^n if there exists scalars x1…xm such that
V = x1v1 + … xm * vm. In other words, a vector is a linear combination of two other vectors if there are scalars x and y such that the vector equals = x * first vector + y * second vector. Add z for a third candidate combined vector, and so forth.
algebraic rules for matrix * vector
M(vector x + vector y) = M(vector x) + M(vector y)
M(k * vector x) = k(M * vector x)
matrix form of a linear system
can write augmented matrix of a linear system [ M | V] as Mv = V. That’s the matrix form of a linear system.
