Transcript 3.2 Subspaces Bases and Linear Independence

3.2 Bases and Linear Independence
Every year
Linear Independence Day
is Celebrated in the US
on July 4
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Independence
A set of vectors is said to be (Linearly)
independent if no vector is a linear
combination of the other vectors.
Problem 10
Determine if the vectors are linearly
independent
Problem 10 Solution
Problem 16
Determine if the vectors are Linearly
Independent or dependent.
Problem 16 solution
What method could we use to
systematically check for
independence?
What method could we use to
systematically check for
independence?
Write the vectors as the columns of a matrix.
Find rref(A)
The columns with leading ones are
independent.
The vectors with out leading ones are
dependent.
Subspace
A subset W in Rn is a subspace if it has the
following 3 properties
a) W contains the zero Vector in Rn
b) W is closed under addition (of two vectors
are in W then their sum is in W)
c) W is closed under scalar multiplication
What are all of the possible vector
subspaces in R2?
What are all of the possible subspaces in
R3?
What are all of the possible vector
subspaces in R2?
The zero vector
Lines passing through the origin
All of R2
What are all of the possible subspaces in
R3?
The zero vector
Lines passing through the origin
Planes passing through the origin
All of R3
Examples of vector Subspaces
A(x) is a linear transformation from Rm to Rn
Is ker(A) a subspace in Rm?
Is the image (A) a subspace in Rn?
Is the line y = x a subspace of R2
Is the line y = x + 1 a subspace of R2
Examples of vector Subspaces
Is the line y = x a subspace of R2 yes
Is the line y = x + 1 a subspace of R2 No
A(x) is a linear transformation from Rm to Rn
Is ker(A) a subspace in Rm? yes
Is the image (A) a subspace in Rn? yes
Basis
A set of vectors is a basis of a subspace if the
vectors are independent and they span the
subspace.
Problem 28
Find a basis of the image (column space)
Problem 28 solution
Example 4
Consider the matrix
Column 2 is a multiple of column 1 and
therefore adds no new information about the
column space (Image)
Column 4 is a sum of columns 1 and 3
therefore it also gives no new information
about the column space (Image)
Example 4
The image of A can be spanned by 2 vectors but not
by 1 alone. (using 3 vectors would be redundant)
Therefore say the column vectors from the first and
third column of the matrix form a basis of the
subspace
Homework p. 121 11-33 odd