Linear Algebra Chapter 1.1 - 1.3 Flashcards
1# Theorem: Let a,b be real numbers. Consider the equation ax=b. What three things are true?
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Define linear algebra
The study of systems of linear equations and their solution sets
2# Theorem: Let a,b,c,d,u,v ∈ R. What is the system?
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Prove 2# theorem
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Prove 1# theorem
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3# definition? Hint: define linear equation
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4# Definition? Hint: Define System of linear equation
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5# definition? Hint: Define solution
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6# definition? hint: solution set
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7# definition? Hint: System equivalent
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8# definition? Hint: consistent
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What part of the matrix is an augmented matrix?
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What part of the matrix is a coefficient matrix?
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What are 6 steps to the general approach to determining solution sets?
- Replace system by an equivalent simpler system
- Subtract second equation by the first
- Subtract third equation by the first
- Subtract third equation by the second (3 times)
- Subtract second equation 3 times by the third equation and subtract the first equation by the third
- Subtract first equation by the second once.
What are 3 operations that simplifies a linear system?
- Replace one equation by the sum of itself and a multiple of another equation
- Interchange two equations
- Multiply equation by a non-zero constant
What is the 3 steps of the elementary row operations?
- Replace one row by the sum of itself and a multiple of another row
- Interchange two rows (swap two rows)
- Multiply all entries in a row by a nonzero constant
What are 3 steps of corresponding row operations?
- Replace one equation by itself plus a multiple of another
- Interchange two equations
- Multiply equation by a nonzero constant
9# definition? Hint: Row equivalent
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10# fact? Hint: two systems and solution sets
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When do we say a matrix is in echelon form? 3 things
- 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
When do we say a matrix is in reduced echelon form? 2 things
- All leading entries are 1 and
- Each leading 1 is the only nonzero entry in its column
Define leading entry
The leading entry in a row is the leftmost nonzero entry in the row
11# theorem?
11# theorem: Every matrix is equivalent to a matrix in echelon form, and to one in reduced echelon form
12# theorem?
12# theorem: Reduced row echelon form is unique: two matrices are equivalent if and only if their reduced echelon forms are the same
What are 4 steps of Row Reduction Algorithm? Forward phase
- Select left most non zero entry. This is a pivot. Swap the rows to put the pivot in the top row.
- Add multiples of the first row to the others to ensure that there are zeros below the pivot
- Ignore the top row and go back to step 1 to select the next pivot
- If no further pivots are available, the matrix is in echelon form
What is 3 steps of Row reduction backward phase?
- Select rightmost pivot
- Add multiples of the pivot row to the rows above it to make the entries in the pivot column equal to 0
- Select the next pivot from the right and repeat step 2.
What are 4 steps to find solution sets of systems of linear equations?
- Suppose you have a linear system whose augmented matrix is in reduced row echelon form
- The system is inconsistent if and only if the augmented column is a pivot column
- Variables corresponding to pivot columns are called basic variables
- Variables corresponding to non-pivot columns are called free variables
Define basic variables
Variables corresponding to pivot columns are called basic variables
Define free variables
Variables corresponding to non-pivot columns are called free variables
13# definition? Hint: Column Vector spaces
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How do we use vector operations geometrically?
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What does a zero vector look like?
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What are 8 algebraic properties of R^n?
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14# definition? Hint: Linear combination
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How do we prove u+v = v+u?
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Define consequence
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15# definition Hint:span
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16# properties Hint: Span
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