The Matrix of a Linear Transformation

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The Matrix of a Linear Transformation
02/28/2008

• We have seen that if
• A is an m ? n matrix
• x is a vector in Rn,
• then the function is a linear transformation
  from Rn to Rm .
• In fact these are the only linear transformations
  from Rn to Rm .

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The Matrix of a Linear Transformation

• Theorem. Let T Rn ? Rm be a linear
  transformation. Then there is a unique matrix A
  with T(x) Ax for all x in Rn.
• The matrix, A, is called the standard matrix for
  T and has the form A T(e1) T(e2) ? ? ?
  T(en)
• where ek (0, ? , 0, 1, 0, ? , 0)
  ? kth coordinate

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The Matrix of a Linear Transformation

• Example. Find the standard matrix A for the
  linear transformation T R3 ? R4 defined by

• What is the size of A?
• What do we need to calculate?

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The Matrix of a Linear Transformation

• Example. Find the standard matrix A for

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The Matrix of a Linear Transformation

• Example. Find the standard matrix A for

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The Matrix of a Linear Transformation

• Example. Find the standard matrix A for

Check
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The Matrix of a Linear Transformation

• Example. Find the standard matrix A for

Do you see an easy way to find A when T is
described this way?
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Linear TransformationsOne-to-One and Onto

• Definitions. A function T Rn ? Rm is
• onto if each b in Rm is the image of at least one
  element of Rn.
• one-to-one (or 1-1) if each b in Rm is the image
  of at most one element of Rn.

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  is
• onto if, given any element b in Rm, there is at
  least one x in Rn with T(x) b.
• one-to-one if, whenever x1 and x2 are any
  elements of Rn with T(x1) T(x2), then x1 x2.

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  is
• onto if, given any element b in Rm, there is at
  least one x in Rn with T(x) b.
• one-to-one if, whenever x1 and x2 are any
  elements of Rn with T(x1) T(x2) 0, thenx1
  x2 0.

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  is
• onto if, given any element b in Rm, there is at
  least one x in Rn with T(x) b.
• one-to-one if, whenever x1 and x2 are any
  elements of Rn with T(x1 x2) 0, thenx1 x2
  0.

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  is
• onto if, given any element b in Rm, there is at
  least one x in Rn with T(x) b.
• one-to-one if, whenever x is any element of Rn
  with T(x) 0, then x 0.
• How can we check if T is 1-1 and/or onto?

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  with standard matrix A is
• onto if, given any element b in Rm, there is at
  least one x in Rn with Ax b.
• one-to-one if, whenever x is any element of Rn
  with Ax 0, then x 0.
• How can we check these? Use what we know about
  matrix equations.

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Linear TransformationsOne-to-One and Onto

• Definitions. A linear transformation T Rn ? Rm
  with standard matrix A is
• onto if a row echelon form of A has a pivot in
  every row (a total of m pivots). (so Ax b
  always solvable)
• one-to-one if a row echelon form of A has a
  pivot in every column (a total of n pivots).
  (so there is a unique solution to Ax 0)

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Linear TransformationsOne-to-One and Onto

• Example. (similar to p. 90 2, 26) Given T R3 ?
  R2 with T(e1) (1, ?4), T(e2) (?2,9) and
  T(e3) (3, ?8).
• Find the standard matrix, A, of T.
• Is T one-to-one?
• Is T onto?
• Procedure. Find A reduce A look for pivot
  positions.
• 1-1 pivot in every column (unique solution to
  Ax 0)
• onto pivot in every row (Ax b is always
  solvable)

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Linear TransformationsOne-to-One and Onto
Example. (similar to p. 90 2, 26) Given T R3 ?
R2 with T(e1) (1, ?4), T(e2) (?2,9) and
T(e3) (3, ?8). Find the standard matrix, A, of
T. Is T 1-1? Is T onto?
Procedure. Find A reduce A look for pivot
positions. 1-1 pivot in every column (unique
soln. to Ax 0) onto pivot in every row (Ax
b always solvable)
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Linear TransformationsOne-to-One and Onto
Example. (similar to p. 90 2, 26) Given T R3 ?
R2 with T(e1) (1, ?4), T(e2) (?2,9) and
T(e3) (3, ?8). Find the standard matrix, A, of
T. Is T 1-1? Is T onto?
Procedure. Find A reduce A look for pivot
positions. 1-1 pivot in every column T is not
1-1 onto pivot in every row T is onto
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Linear TransformationsOne-to-One and Onto

• Example. (similar to p. 91 19, 27) Given T R3
  ? R4 where
• T(x1, x2, x3) (x1 3x2, 2x2 3x3, 5x3, 0).
  
• Find the standard matrix, A, of T. Is T 1-1?
  Is T onto?
• Procedure. Find A reduce A look for pivot
  positions.

Answer
but T is not onto
T is 1-1
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