Linear Algebra Chapter 1.1 - 1.3

Question 1

1# Theorem: Let a,b be real numbers. Consider the equation ax=b. What three things are true?

Answer 1

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Question 2

Define linear algebra

Answer 2

The study of systems of linear equations and their solution sets

Question 3

2# Theorem: Let a,b,c,d,u,v ∈ R. What is the system?

Answer 3

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Question 4

Prove 2# theorem

Answer 4

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Question 5

Prove 1# theorem

Answer 5

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Question 6

3# definition? Hint: define linear equation

Answer 6

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Question 7

4# Definition? Hint: Define System of linear equation

Answer 7

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Question 8

5# definition? Hint: Define solution

Answer 8

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Question 9

6# definition? hint: solution set

Answer 9

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Question 10

7# definition? Hint: System equivalent

Answer 10

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Question 11

8# definition? Hint: consistent

Answer 11

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Question 12

What part of the matrix is an augmented matrix?

Answer 12

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Question 13

What part of the matrix is a coefficient matrix?

Answer 13

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Question 14

What are 6 steps to the general approach to determining solution sets?

Answer 14

1. Replace system by an equivalent simpler system
2. Subtract second equation by the first
3. Subtract third equation by the first
4. Subtract third equation by the second (3 times)
5. Subtract second equation 3 times by the third equation and subtract the first equation by the third
6. Subtract first equation by the second once.

Question 15

What are 3 operations that simplifies a linear system?

Answer 15

1. Replace one equation by the sum of itself and a multiple of another equation
2. Interchange two equations
3. Multiply equation by a non-zero constant

Question 16

What is the 3 steps of the elementary row operations?

Answer 16

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

Question 17

What are 3 steps of corresponding row operations?

Answer 17

1. Replace one equation by itself plus a multiple of another
2. Interchange two equations
3. Multiply equation by a nonzero constant

Question 18

9# definition? Hint: Row equivalent

Answer 18

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Question 19

10# fact? Hint: two systems and solution sets

Answer 19

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Question 20

When do we say a matrix is in echelon form? 3 things

Answer 20

• Any zero row occurs below any nonzero row, and
• The leading entry in any nonzero row occurs in a column strictly to the right of the leading entries above it
• All entries in a column below a leading entry are zeros

Question 21

When do we say a matrix is in reduced echelon form? 2 things

Answer 21

• All leading entries are 1 and

- Each leading 1 is the only nonzero entry in its column

Question 22

Define leading entry

Answer 22

The leading entry in a row is the leftmost nonzero entry in the row

Question 23

11# theorem?

Answer 23

11# theorem: Every matrix is equivalent to a matrix in echelon form, and to one in reduced echelon form

Question 24

12# theorem?

Answer 24

12# theorem: Reduced row echelon form is unique: two matrices are equivalent if and only if their reduced echelon forms are the same

Question 25

What are 4 steps of Row Reduction Algorithm? Forward phase

Answer 25

1. Select left most non zero entry. This is a pivot. Swap the rows to put the pivot in the top row.
2. Add multiples of the first row to the others to ensure that there are zeros below the pivot
3. Ignore the top row and go back to step 1 to select the next pivot
4. If no further pivots are available, the matrix is in echelon form

Question 26

What is 3 steps of Row reduction backward phase?

Answer 26

1. Select rightmost pivot
2. Add multiples of the pivot row to the rows above it to make the entries in the pivot column equal to 0
3. Select the next pivot from the right and repeat step 2.

Question 27

What are 4 steps to find solution sets of systems of linear equations?

Answer 27

1. Suppose you have a linear system whose augmented matrix is in reduced row echelon form
2. The system is inconsistent if and only if the augmented column is a pivot column
3. Variables corresponding to pivot columns are called basic variables
4. Variables corresponding to non-pivot columns are called free variables

Question 28

Define basic variables

Answer 28

Variables corresponding to pivot columns are called basic variables

Question 29

Define free variables

Answer 29

Variables corresponding to non-pivot columns are called free variables

Question 30

13# definition? Hint: Column Vector spaces

Answer 30

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Question 31

How do we use vector operations geometrically?

Answer 31

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Question 32

What does a zero vector look like?

Answer 32

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Question 33

What are 8 algebraic properties of R^n?

Answer 33

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Question 34

14# definition? Hint: Linear combination

Answer 34

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Question 35

How do we prove u+v = v+u?

Answer 35

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Question 36

Define consequence

Answer 36

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Question 37

15# definition Hint:span

Answer 37

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Question 38

16# properties Hint: Span

Answer 38

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