Linear Algebra and Its Applications
Math 220: Linear Algebra, Section 1.9 Haijun Li Department of Mathematics Washington State University
Spring 2012
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
1 / 10
Linear Algebra and Its Applications
Recall
A mapping T : Rn → Rm is called a linear transformation with domain Rn and codomain Rm if T(cu + dv) = c T(u) + d T(v) for all vectors u, v and all scalars c, d.
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
2 / 10
Linear Algebra and Its Applications
Recall
A mapping T : Rn → Rm is called a linear transformation with domain Rn and codomain Rm if T(cu + dv) = c T(u) + d T(v) for all vectors u, v and all scalars c, d. Example: If A is an m × n matrix, then T(x) := Ax is a linear transform from Rn to Rm .
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
2 / 10
Linear Algebra and Its Applications
Recall
A mapping T : Rn → Rm is called a linear transformation with domain Rn and codomain Rm if T(cu + dv) = c T(u) + d T(v) for all vectors u, v and all scalars c, d. Example: If A is an m × n matrix, then T(x) := Ax is a linear transform from Rn to Rm . That is, a matrix determines a linear transform.
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
2 / 10
Linear Algebra and Its Applications
Recall
A mapping T : Rn → Rm is called a linear transformation with domain Rn and codomain Rm if T(cu + dv) = c T(u) + d T(v) for all vectors u, v and all scalars c, d. Example: If A is an m × n matrix, then T(x) := Ax is a linear transform from Rn to Rm . That is, a matrix determines a linear transform. Conversely, any linear transform T : Rn → Rm uniquely determines a matrix.
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
2 / 10
Linear Algebra and Its Applications
Identity Matrix The identity matrix In is an n × n matrix with 1’s on the main diagonal and 0’s elsewhere. That is, 1 0 ··· 0 0 1 ··· 0 A= . . . . . ... .. .. 0 0 ···
Haijun Li
Math 220: Linear Algebra, Section 1.9
1
Spring 2012
3 / 10
Linear Algebra and Its Applications
Identity Matrix The identity matrix In is an n × n matrix with 1’s on the main diagonal and 0’s elsewhere. That is, 1 0 ··· 0 0 1 ··· 0 A= . . . . . ... .. .. 0 0 ···
1
Let ej denote the j-th column vector of In .
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
3 / 10
Linear Algebra and Its Applications
Identity Matrix The identity matrix In is an n × n matrix with 1’s on the main diagonal and 0’s elsewhere. That is, 1 0 ··· 0 0 1 ··· 0 A= . . . . . ... .. .. 0 0 ···
1
Let ej denote the j-th column vector of In . Example: Let
1 0 0 I3 = 0 1 0 = [e1 , e2 , e3 ]. 0 0 1 Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
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Linear Algebra and Its Applications
Example For any x in R3 , x1 x = I3 x = [e1 , e2 , e3 ] x2 = x1 e1 + x2 e2 + x3 e3 . x3
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
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Linear Algebra and Its Applications
Example For any x in R3 , x1 x = I3 x = [e1 , e2 , e3 ] x2 = x1 e1 + x2 e2 + x3 e3 . x3
Let T : R3 → R2 be a linear transform, then T(x) = T(x1 e1 + x2 e2 + x3 e3 ) = x1 T(e1 ) + x2 T(e2 ) + x3 T(e3 ) x1 x1 = [T(e1 ), T(e2 ), T(e3 )] x2 = A x2 , x3 x3 where A = [T(e1 ), T(e2 ), T(e3 )] is a 2 × 3 matrix. Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
4 / 10
Linear Algebra and Its Applications
Example For any x in R3 , x1 x = I3 x = [e1 , e2 , e3 ] x2 = x1 e1 + x2 e2 + x3 e3 . x3
Let T : R3 → R2 be a linear transform, then T(x) = T(x1 e1 + x2 e2 + x3 e3 ) = x1 T(e1 ) + x2 T(e2 ) + x3 T(e3 ) x1 x1 = [T(e1 ), T(e2 ), T(e3 )] x2 = A x2 , x3 x3 where A = [T(e1 ), T(e2 ), T(e3 )] is a 2 × 3 matrix. That is, the transform T is determined by A. Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
4 / 10
Linear Algebra and Its Applications
For any x in Rn , x1 x = In x = [e1 , · · · , en ] ... = x1 e1 + x2 e2 + · · · + xn en . xn
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
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Linear Algebra and Its Applications
For any x in Rn , x1 x = In x = [e1 , · · · , en ] ... = x1 e1 + x2 e2 + · · · + xn en . xn
Let T : Rn → Rm be a linear transform, then T(x) = T(x1 e1 +x2 e2 +· · ·+xn en ) = x1 T(e1 )+x2 T(e2 )+· · ·+xn T(en ) x1 x1 = [T(e1 ), T(e2 ), · · · , T(en )] ... = A ... , xn
xn
where A = [T(e1 ), T(e2 ), · · · , T(en )], called the standard matrix for the linear transformation T, is an m × n matrix. Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
5 / 10
Linear Algebra and Its Applications
Example Let T : R2 → R3 be a linear transform defined by x1 − 2x2 . 4x1 T(x) = 3x1 + 2x2 Find the standard matrix for the linear transformation T.
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
6 / 10
Linear Algebra and Its Applications
Example Let T : R2 → R3 be a linear transform defined by x1 − 2x2 . 4x1 T(x) = 3x1 + 2x2 Find the standard matrix for the linear transformation T. Solution: Since 1 −2 T(e1 ) = 4 , T(e2 ) = 0 , 3 2
1 −2 [T(e1 ), T(e2 )] = 4 0 . 3 2 Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
6 / 10
Linear Algebra and Its Applications
Example
Find the standard matrix for the contraction/dilation transformation T(x) = k x for x in R2 .
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
7 / 10
Linear Algebra and Its Applications
Example
Find the standard matrix for the contraction/dilation transformation T(x) = k x for x in R2 . Solution: Since k 0 T(e1 ) = k e1 = , T(e2 ) = k e2 = , 0 k [T(e1 ), T(e2 )] =
Haijun Li
k 0 0 k
Math 220: Linear Algebra, Section 1.9
.
Spring 2012
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Linear Algebra and Its Applications
Shears The standard matrix for the shear transformation x x + ky T( )= for x in R2 is given by y y 1 k image147.gif (GIF Image, 325 × 125 pixels) [T(e1 ), T(e2 )] = . 0 1
Haijun Li
Math 220: Linear Algebra, Section 1.9
h
Spring 2012
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Linear Algebra and Its Applications
One-One and Onto Definition A linear transform T : Rn → Rm is onto if each b in codomain Rm is the image of at least one x in domain Rn .
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
9 / 10
Linear Algebra and Its Applications
One-One and Onto Definition A linear transform T : Rn → Rm is onto if each b in codomain Rm is the image of at least one x in domain Rn . Definition A linear transform T : Rn → Rm is one-to-one if each b in codomain Rm is the image of at most one x in domain Rn . ILT.png (PNG Image, 294 × 196 pixels)
Haijun Li
Math 220: Linear Algebra, Section 1.9
http://www.aimath.org
Spring 2012
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Linear Algebra and Its Applications
Theorem Let T : Rn → Rm be a linear transform with standard matrix A.
Haijun Li
1
T is onto if and only if the columns of A spans Rm .
2
T is one-to-one if and only if the columns of A are linearly independent.
Math 220: Linear Algebra, Section 1.9
Spring 2012
10 / 10
Linear Algebra and Its Applications
Theorem Let T : Rn → Rm be a linear transform with standard matrix A. 1
T is onto if and only if the columns of A spans Rm .
2
T is one-to-one if and only if the columns of A are linearly independent.
Example: Let T : R2 → R3 be a linear transform defined by x1 − 2x2 . 4x1 T(x) = 3x1 + 2x2 Since the columns of the standard matrix are linearly independent, T is one-to-one (but not onto).
Haijun Li
Math 220: Linear Algebra, Section 1.9
Spring 2012
10 / 10
