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Chapter 3 Vector Spaces
Definition and Examples
Subspaces
Linear Independence
Basis and Dimension
Change of Basis
Row Space and Column Space
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1. Definition and Examples
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1. Definition and Examples
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1. Definition and Examples
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2. Subspaces
Ex. 1: Let
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2
1xx
x
xS
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2. Subspaces
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2. Subspaces
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2. Subspaces
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2. Subspaces
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2. Subspaces
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2. Subspaces– Span of a Set of
Vectors
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2. Subspaces– Span of a Set of
Vectors
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2. Subspaces– Span of a Set of
Vectors
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3. Linear Independence
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3. Linear Independence
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3. Linear Independence
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3. Linear Independence
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3. Linear Independence
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3. Linear Independence
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3. Linear Independence
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4. Basis and Dimension
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4. Basis and Dimension
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4. Basis and Dimension
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4. Basis and Dimension
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4. Basis and Dimension
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4. Basis and Dimension
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4. Basis and Dimension
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5. Change of Basis
Changing Coordinates in R2
Standard basis is {e1, e2}
Scalars x1 and x2 are the coordinates of x with respect to the standard basis.
For any basis {y, z} for R2
Scalars and are the coordinates of x with respect to the basis [y, z] .
The vector ( )T is the coordinate vector of xwith respect to [y, z]
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5. Change of Basis (c.1)
Coordinate vector w.r.p. to [y, z]
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5. Change of Basis– Changing
Coordinates
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5. Change of Basis– Changing
Coordinates
Transition Matrix from ordered basis [u1, u2] to ther basis [e1, e2]
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5. Change of Basis– Changing
Coordinates
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5. Change of Basis– Changing
Coordinates
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5. Change of Basis– Changing
Coordinates
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5. Change of Basis– Changing
Coordinates
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5. Change of Basis– Change of Basis
for a General Vector Space
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5. Change of Basis– Change of Basis
for a General Vector Space
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5. Change of Basis– Change of Basis
for a General Vector Space
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5. Change of Basis– Change of Basis
for a General Vector Space
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5. Change of Basis– Change of Basis
for a General Vector Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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6. Row Space and Column Space
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