Chapter 1

Question 1

What are the elementary row operations?

Answer 1

(Replacement) Replace one row by the sum of itself and a multiple of another row.
(Interchange) Interchange two rows.(Scaling) Multiply all entries in a row by a nonzero constant.

Question 2

How do we know if a linear system is consistent or inconsistent?

Answer 2

A consistent linear system has at least one solution while an inconsistent linear system has no solution.

Question 3

what is an m x n matrix?

Answer 3

a matrix with m rows and n columns.

Question 4

coeffiecent vs augmented matrix

Answer 4

coeffiecent matrixs are just the coeffecients of the variables while augmented matrix have the constants on the otherside of the equal sign. (I don’t wanna say solutions b/c I don’t wanna confuse terminology)

Question 5

what makes a equation linear?

Answer 5

every variable is to the first power. you can have subscripts. No variables are multiplied.

Question 6

Roughly speaking, how do you solve linear systems?

Answer 6

use the x_1term in the first equation of a system to eliminate the x_1terms in the other equations. Then use the x_2term in the second equation to eliminate the x_2terms in the other equations, and so on, until you finally obtain a very simple equivalent system of equations.
*Get it in eichon form. With the staircase of 0’s

Question 7

graphically, what do consistent and inconsistent systems look like?

Answer 7

Inconsistent would be two parallel lines. Consistent systems intersect at at least one point. -> could even intersect the entire lines which would be infinitely many solutions.

Question 8

echoleon form vs row reduced echoleon form

Answer 8

Echelon form is where there are zeros below each leading entry. row reduced echeleon form is where every leading entry is 1 and every other entry is a 0 or the solution to the variables.

Question 9

Let A be an m x nmatrix. Then the following statements are logically equivalent. That is, for a particular b, either they are all true statements or they are all false.

Answer 9

• For each b in R^m, the equation Ax=bhas a solution.
• Each b in R^mis a linear combination of the columns of .
• The columns ofA span R^m.
• A has a pivot position in every row.

Question 10

what are the components of Ax=b

Answer 10

A is the matrix of numbers
x is the variables in R^m
b is the solution of the set

Question 11

how does matrix multiplication work?

Answer 11

Think dot product. Make a column vector then first entry of column vector by first entries of matrix, 2nd for 2nd and so on.

Question 12

what are homogeneous linear systems and equations

Answer 12

homogenuous equation is of the form Ax = 0. so a a homogeneous system is a system of these.

Question 13

trivial vs non trivial solution

Answer 13

trivial - the of solution of x =0 to a homogeneous eq
non - trivial is a non zero solution

Question 14

when does a homogeneous eq have a nontrivial solution?

Answer 14

iff the EQ has at least one free variable.

Question 15

parametric form of vector equation

Answer 15

x = a + tb
a and b are vectors

Question 16

how to know vectors are linearly independent

Answer 16

• it has only the trivial solution
• not scalar multiples of each other
• if they are in the span of each other

Question 17

how to know if vectors are linearly dependent.

Answer 17

• there exist weights not all zero. The vector equation has non trivial solutions
• free variable
• scalar multiples of each other
• vectors > entries (rows > columns)

Question 18

what is Span {u,v} actually?

Answer 18

it’s the set of linear combinations of vectors. particularly within a plane.

Question 19

when is An indexed set S = {v_1,….,v_p}of two or more vectors is linearly dependent?

Answer 19

if and only if at least one of the vectors in S is a linear combination of the others. In fact, if Sis linearly dependent and V_1 doesn’t = 0,then some v_j(with j>1) is a linear combination of the preceding vectors, v1, …, v_j-1.

Question 20

whats a linear combination?

Answer 20

A linear combination of vectors is created by scaling each vector (adjusting its quantity) and then adding them together.