线性代数 英文讲义

be ordered bases for Rn and Rm, respectively. If L: Rn → Rm is a linear transformation and A is the matrix representing L with respect to E and F, then aj=B-1L(uj) where B=(b1, …,bm). for j=1, 2, …, n
2 Matrix Representations of Linear Transformation
Theorem 4.2.1 If L is a linear transformation mapping Rn
into Rm, there is an m×n matrix A such that L(x)=Ax for each x Rn. In fact, the jth column vector of A is given x∈R by aj=L(ej) j=1, 2, …, n
B=U-1AU
Theorem 4.3.1 Let E=[v1, v2, …,vn] and F=[w1,w2, … ,wn] be
two ordered bases for a vector space V, and let L be a linear operator on V. Let S be the transition matrix representing the change from F to E. If A is the matrix representing L with respect to E, and B is the matrix representing L with respect to F, then B=S-1AS.
1 2 1 2
2 b1=U-1L(u1)= 0
1 b2=U-1L(u2)= 1
2 1 B= 0 1
Thus, if (1) B is the matrix representing L with respect to [u1, u2] (2) A is the matrix representing L with respect to [e1, e2] (3) U is the transition matrix corresponding to the change of basis from [u1, u2] to [e1, e2] then
L=LA
v∈V
w=L(v) ∈W
x=[v]E
∈Rn
A
Ax=[w]F∈Rm
Example
Let L be a linear transformation mapping R3
into R2 defined by L(x)=x1b1+(x2+x3)b2 for each x∈R3,where
1 b1 = 1
2 Similarity
If L is a linear operator on an n-dimensional vector space V, the matrix representation of L will depend on the ordered basis chosen for V. By using different bases, it is possible to represent L by different n×n matrices.
for each x∈R2.
Example
Let L be the operator defined by L(x)=(x1,-x2)T
for each x=(x1,x2)T∈R2.
Linear Transformations from Rn to Rm
Example
The mapping L: R2→R1 defined by L(x)=x1+x2
Theorem 4.1.1 If L: V →W is a linear transformation
and S is a subspace of V, then (1) Ker(L) is a subspace of V. (2) L(S) is a subspace of W.
Example
Let L be the linear operator on R2 defined by
x1 L( x ) = 0
Example
Let L: R3→R2 be tຫໍສະໝຸດ Baidue linear transformation
defined by L(x)=(x1+x2, x2+x3)T and let S be the subspace of R3 spanned by e1 and e3.
Linear Transformations from V to W
If L is a linear transformation mapping a vector space V into a vector space W, then (1) L(0v)=0w (where 0v and 0w are the zero vectors in V and W, respectively). (2) If v1, …, vn are elements of V and α1, …, αn are scalars, then L(α1v1+ α2v2+‥‥+ αnvn)= α1L(v1)+ α2L(v2）+‥‥+ αnL(vn) (3) L(-v)=-L(v) for all v∈V.
A is the matrix representing L relative to the ordered bases
If A is the matrix representing L with respect to the bases E and F and x=[v]E (the coordinate vector of v with respect to E) y=[w]F (the coordinate vector of w with respect to F) then L maps v into w if and only if A maps x into y.
is a linear transformation.
Example
Consider the mapping M defined by
M (x) = ( x + x )
2 1 2 2
1
2
Example
The mapping L from R2 to R3 defined by
L(x)=(x2 , x1, x1+x2)T is a linear transformation.
Chapter 4
Linear Transformations
1 Definition and Examples
Definition
A mapping L from a vector space V into a vector space W is said to be a linear transformation if (1) L( α v1+βv2)= α L(v1)+βL(v2)
Corollary 4.2.4 If A is the matrix representing the linear
transformation L: Rn→Rm with respect to the bases E=[u1, u2, …,un] and F=[b1,b2, … ,bm] then the reduced row echelon form of (b1, …,bm ∣L(u1), …,L(un)) is (I∣A).
If there is another basis for
R2:
1 1 u1 = ， u 2 = 1 1
2 0 1 2 L(u1)=Au1= 1 1 1 = 2
2 0 1 2 L(u2)=Au2= 1 1 1 = 2
for all v1, v2 ∈V and for all scalars α and β.
Linear Operators on R2
Example
Let L be the operator defined by L(x)=3x
for each x∈R2.
Example
Consider the mapping L defined by L(x)=x1e1
Example
defined by
Let L: R2→R3 be the linear transformation
L(x)=(x2, x1+x2, x1-x2)T Find the matrix representations of L with respect to the ordered bases [u1, u2] and [b1, b2, b3], where u1=(1, 2)T, and b1=(1, 0, 0)T, b2=(1, 1, 0)T, b3=(1, 1, 1)T u2=(3, 1)T
Example
Define the linear transformation L: R3→R2 by L(x)=(x1+x2, x2+x3)T
Theorem 4.2.2 (Matrix Representation Theorem)
If E=[v1, v2, …,vn] and F=[w1,w2, … ,wm] are ordered bases for vector spaces V and W, respectively, then corresponding to each linear transformation L: V → W there is an m×n matrix A such that [L(v)]F=A[v]E E and F. In fact, aj=[L(vj)]F j=1, 2, …, n for each v∈V
1 1 The transition matrix from [u1, u2] to [e1, e2] is U=(u1, u2)= 1 1
1 The transition matrix from [e1, e2] to [u1, u2] is U-1= 2 1 2
Let matrix B representing L with repect to [u1, u2],
Example
defined by
Let L: R2→R2 be the linear transformation
L(x)=(2x1, x1+x2)T
So the matrix representing L with respect to [e1, e2] is A = 1 1 2 0
and
1 b2 = 1
Find the matrix A representing L with respect to the ordered bases [e1, e2, e3] and [b1, b2].
Theorem 4.2.3 Let E=[u1, u2, …,un] and F=[b1,b2, … ,bm]
The Image and Kernel
Definition
Let L: V→W be a linear transformation. The kernel of L, denoted Ker(L), is defined by Ker(L)={ v∈V ∣L(v)=0w}
Definition
Let L: V→W be a linear transformation and let S be a subspace of V. The image of S, denoted L(S), is defined by L(S)={ w∈W ∣w=L(v) for some v∈ S} The image of the entire vector space, L(V), is called the range of L.