3. This example hopes to illustrate why the vector spaces the linear transformation are defined o...
How was the linear transformation of b1 and b2 were applied (L(b1) , L(b2))? NOTE: b1=(1,1)^T , b2=(-1,1)^T Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R? into itself and defined by where (bi, b2] is the ordered basis defined in Example 3. Find the matrix A represent- ing L with respect to [bi, b2l Solution Thus, A0 2 onofosmation D defined by D(n n' maps P into P, Given the ordered Linear Transformations EXAMPLE 4 Let...
4) The linear transformation L defined by L(p(x)) = p'(x)+p(0) maps Pinto P. a) Find the matrix representation of L with respect to the ordered bases {1,x,x} and {1, 1-x}. 6 b) For the vector, p(x) = 2x + x - 2 (i) find the coordinates of L(p(x)) with respect to the ordered basis{1, 1-x}. , using the matrix you found in a). Remember to use the coordinate vector of p(x) with respect to the basis {1,x,x"}. (ii) Show that...
4) The linear transformation L defined by L(p(x)) = p'(x)+ p(0) maps P, into P. a) Find the matrix representation of L with respect to the ordered bases {1xx.x"} and {1, 1-x} b) For the vector, p(x) = 2x2 + x-2 () find the coordinates of L(p(x)) with respect to the ordered basis {1, 1-x}., using the matrix you found in a). Remember to use the coordinate vector of p(x) with respect to the basis {1xx"}. (ii) Show that they...
4) The linear transformation L defined by L(p(x)) = p(x)+p(0) maps Pinto P. a) Find the matrix representation of L with respect to the ordered bases l_r"} and {1, 1-x). b) For the vector, p(x) = 2x' +1-2 () find the coordinates of L(p(x)) with respect to the ordered basis{1, 1-x), using the matrix you found in a). Remember to use the coordinate vector of p(x) with respect to the basis {1x2). (ii) Show that they are the weights that...
Font Styles Paragraph Definition 1: Given La linear transformation from a vector space V into itself, we say that is diagonalizable iff there exists a basis S relevant to which can be represented by a diagonal matrix D. Definition 2: If the matrix A represents the linear transformation L with respect to the basis S, then the eigenvalues of L are the eigenvalues of the matrix A. I Definition 3: If the matrix A represents the linear transformation L with...
Let T R3 R4 be the linear transformation defined by T(π1, Ο2, 73) - ( 3α1 -4 , X3, 12.x2 3.x3, 6x1-25x3, 10x2 + 10x3) (a) Determine the standard matrix representation of T (b) Find a basis for the image of T, Im(T), and determine dim(Im(T)) (c) Find a basis for the kernel of T, ker(T), and determine dim(ker(T))
11. =(7.5), #,(-3,-1) 2) Let = (1.-5). v. =(-2,2) and let L be a linear operator on R whose matrix representation with respect to the ordered basis . is a) Determine the transition matrix (change of basis matrix) from, v,to (1) (Draw the commutative triangle). 3 b) Find the matrix representation B, of L with respect to ,v} by USING the similarity relation
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...
1 6) Let L: R→ R* be defined as L(A) = A. (1 2) (1996.)A OC :) The standard basis for R2 is E = { Find the matrix representation of L with respect to E. (Hint: the matrix that represents the linear transformation, in this case, must be 4x4)
Q4. Let L: R2 + Rº be a transformation defined by L (0-2 [3u2 – U1 U1 – U2 -502 (a) Show that I is a linear transformation. (b) Find the standard matrix A of L, and find L ([31]) using the matrix A. (c) Do you think that any transformation T:R2 + R² is linear? (Justify your answer).