announcements feb 4 - peoplepeople.math.gatech.edu/.../slides/feb10.pdf · announcements feb 4 •...
TRANSCRIPT
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Announcements Feb 4
• Midterm 2 on March 6
• WeBWorK 2.6 due Thursday
• My o�ce hours Monday 3-4 and Wed 2-3
• TA o�ce hours in Skiles 230 (you can go to any of these!)
I Isabella Thu 2-3I Kyle Thu 1-3I Kalen Mon/Wed 1-1:50I Sidhanth Tue 10:45-11:45
• PLUS sessions Mon/Wed 6-7 LLC West with Miguel (di↵erent this week)
• Supplemental problems and practice exams on the master web site
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5. Find the reduced row echelon form of the following matrix. Show your work.
0
@0 0 1 2
1 3 �2 1
2 6 0 10
1
A
6. Suppose that there is a matrix equaion Ax = b and that the reduced row echelon form of
the augmented matrix (A|b) is 0
@0 1 �3 0 7
0 0 0 1 2
0 0 0 0 0
1
A
Write the parametric vector form of the solution to Ax = b.
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7. The following diagram indicates tra�c flow in the town square (the numbers indicate the
number of cars per minute on each section of road).
10
1010
10
x1
x2
x3
15
Write down a vector equation describing the flow of tra�c. Do not solve.
8. Find all values of h so that the vectors
⇣11�9
⌘,
⇣016
⌘, and
⇣1hh
⌘are linearly dependent.
Show your work.
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Section 2.7Bases
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Section 2.7 Summary
• A basis for a subspace V is a set of vectors {v1, v2, . . . , vk} such that
1. V = Span{v1, . . . , vk}2. v1, . . . , vk are linearly independent
• The number of vectors in a basis for a subspace is the dimension.
• Find a basis for Nul(A) by solving Ax = 0 in vector parametric form
• Find a basis for Col(A) by taking pivot columns of A (not reduced A)
• Basis Theorem. Suppose V is a k-dimensional subspace of Rn. Then
I Any k linearly independent vectors in V form a basis for V .I Any k vectors in V that span V form a basis.
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Section 2.9The rank theorem
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Rank Theorem
On the left are solutions to Ax = 0, on the right is Col(A):
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Rank Theorem
rank(A) = dimCol(A) = # pivot columns
nullity(A) = dimNul(A) = # nonpivot columns
Rank-Nullity Theorem. rank(A) + nullity(A) = #cols(A)
This ties together everything in the whole chapter: rank A describes the b’s sothat Ax = b is consistent and the nullity describes the solutions to Ax = 0. Somore flexibility with b means less flexibility with x, and vice versa.
Example. A =
0
@1 1 11 1 11 1 1
1
A
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Section 2.9 Summary
• Rank Theorem. rank(A) + dimNul(A) = #cols(A)