Lecture 8: The matrix of a linear transformation. Applications

96 Pages·2014·0.5 MB·English

Lecture 8: The matrix of a linear transformation.
Applications
Danny W. Crytser
April 7, 2014
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R .
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2.
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
Example
2 3
Let T : R → R be the linear transformation defined by
 
([ ]) x
1 + x2
T x1 =  2x1  .
x2
3x2
2
Let’s find the matrix A such that T(x) = Ax for all x ∈ R . Note
2 3
that A must have 2 columns (domain R ) and 3 rows (domain R ).
The two basis vectors in the domain are e1, e2. Their images are
   
1 1
T(e1) =  2  , T(e2) =  0  .
0 3
So A has those vectors as columns
 
1 1
A =  2 0  .
0 3
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
     
[ ] 1 1 x
1 + x2 ([ ])
x1 x1
     
A = x1 2 + x2 0 = 2x1 = T .
x2 x2
0 3 3x2
2
Thus T(x) = Ax for all x ∈ R . There’s a fancy term for the
matrix we’ve cooked up.
Definition
n m
If T : R → R is a linear transformation and e1, e2, . . . , en are
n
the standard basis vectors in R , then the matrix
[ ]
A = T(e1) T(e2) . . . T (en)
n
which satisfies T(x) = Ax for all x ∈ R is called the standard
matrix for T .
We can check to see we got the right answer:
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications
     
1 1 x1 + x2 ([ ])
x1
     
= x1 2 + x2 0 = 2x1 = T .
x2
0 3 3x2
2
Thus T(x) = Ax for all x ∈ R . There’s a fancy term for the
matrix we’ve cooked up.
Definition
n m
If T : R → R is a linear transformation and e1, e2, . . . , en are
n
the standard basis vectors in R , then the matrix
[ ]
A = T(e1) T(e2) . . . T (en)
n
which satisfies T(x) = Ax for all x ∈ R is called the standard
matrix for T .
We can check to see we got the right answer:
[ ]
x1
A
x2
Dan Crytser Lecture 8: The matrix of a linear transformation. Applications