Homework Ch 4.2 Quizlet
Let L: V--> W be a linear transformation between 2 vector spaces. How do we represent L by matrix notation?
-Suppose E= {v1, v2, ..., vn} is a basis of V -Suppose F= {w1, w2, ..., wn} is a basis of W -For any v e V we write v= c1v1 +c2v2 +...+cnVn (linear combo) -We write [V]E = (c1, c2, ... cn)^T which is called the coordinate of v w/ respect to the basis E, which is comprised of the coefficients of the linear combination -For any w e W we write w= a1w1 + a2w2 + ... + amWm -We write [W]F = (a1 + a2+ ... + am)^T which is called the coordinate w/ respect to the basis F - There exists an mxn matrix such that [L(V)]F = A [V]E Note: [L(V)]F is the coordinate of L(V) w/ respect to F and A is the matrix representation of L
4. Let L be the linear operator on R3 defined by L(x)= (2x1-x2-x3) (2x2-x1-x3) (2x3-x1-x2) Determine the standard matrix representation A of L, and use A to find L (x) for each of the following vectors x: a. x = (1, 1, 1)T
Find standard matrix representation A of L: ( 2 -1 -1 ) (-1 2 -1 ) (-1 -1 2)
4. Let L be the linear operator on R3 defined by L(x)= (2x1-x2-x3) (2x2-x1-x3) (2x3-x1-x2) Determine the standard matrix representation A of L, and use A to find L (x) for each of the following vectors x: b. x = (2, 1, 1)T
Find standard matrix representation A of L: ( 2 -1 -1 ) (-1 2 -1 ) (-1 -1 2)
4. Let L be the linear operator on R3 defined by L(x)= (2x1-x2-x3) (2x2-x1-x3) (2x3-x1-x2) Determine the standard matrix representation A of L, and use A to find L (x) for each of the following vectors x: c. x = (−5, 3, 2)T
Find standard matrix representation A of L: ( 2 -1 -1 ) (-1 2 -1 ) (-1 -1 2)
Let L be a linear transformation from R^n to R^m, then there exists an m x n matrix A such that...
L(x)= Ax for any x e R^n to find A, we solve for L(ei)=A(ei) where ei is a unit vector Thus L(ei) is the column I of matrix A
14. The linear transformation L defined by L (p(x)) = p'(x) + p(0) maps P3 into P2. Find the matrix representation of L with respect to the ordered bases [x2, x, 1] and [2, 1 − x]. For each of the following vectors p(x) in P3, find the coordinates of L (p(x)) with respect to the ordered basis [2, 1 − x] b. b. x^2 + 1
Matrix representation of L with respect to ordered basis: A= (1 (1/2) (1/2)) (-2 0 0 )
14. The linear transformation L defined by L (p(x)) = p'(x) + p(0) maps P3 into P2. Find the matrix representation of L with respect to the ordered bases [x2, x, 1] and [2, 1 − x]. For each of the following vectors p(x) in P3, find the coordinates of L (p(x)) with respect to the ordered basis [2, 1 − x] c. 3x
Matrix representation of L with respect to ordered basis: A= (1 (1/2) (1/2)) (-2 0 0 )
14. The linear transformation L defined by L (p(x)) = p'(x) + p(0) maps P3 into P2. Find the matrix representation of L with respect to the ordered bases [x2, x, 1] and [2, 1 − x]. For each of the following vectors p(x) in P3, find the coordinates of L (p(x)) with respect to the ordered basis [2, 1 − x] d. 4x^2 + 2x
Matrix representation of L with respect to ordered basis: A= (1 (1/2) (1/2)) (-2 0 0 )
14. The linear transformation L defined by L (p(x)) = p'(x) + p(0) maps P3 into P2. Find the matrix representation of L with respect to the ordered bases [x2, x, 1] and [2, 1 − x]. For each of the following vectors p(x) in P3, find the coordinates of L (p(x)) with respect to the ordered basis [2, 1 − x] a. x^2 + 2x − 3
Objective: Find matrix A representing L with respect to the standard basis [x^2, x, 1] and [2, 1-x] STEP 1: find L(x^2), L(x) and L(1) which is 2x, 1, and 1 respectively (ex// L(x^2)=p'(x^2)+ p(0) = 2x) STEP 2: Express L(x^2), L(x) and L(1) as a linear combination in terms of the basis [2, 1-x] --> a(2)+ b(1-x) --> a2 + b - bx = L(Plug in values from Step 1 here) STEP 3: Plug in found valued from Step 1 and solve for a and b - L(x^2) = 2x = a2 + b - bx What coefficients will make this work? a=1 and b=-2 -L(x) = 1= a2 +b- bx What coefficients will make this work? a= 1/2 and b= 0 -L(1) = 1= a2 +b- bx What coefficients will make this work? a= 1/2 and b= 0 STEP 4: Create the translation matrix using the points we found. STEP 5: Find the coordinates p(x) with respect to basis: 1, 2, -3. STEP 6: Multiply by matrix to solve
13. Let L be the linear transformation mapping P2 into R2 defined by L (p(x)) = ⎧(integral from 0 to 1)p(x)dx⎫ ⎩ p(0) ⎭ Find a matrix A such that L (α + βx) = A ( α β )^T
Step 1: Perform the linear transformation by plugging in (α + βx) for p(x) in L(p(x)). This will leave us with our left hand side. Step 2: Put together the matrix you found for (L(p(x)) = A(α β )^T. Solve for matrix A
