11. =(7.5), #,(-3,-1) 2) Let = (1.-5). v. =(-2,2) and let L be a linear operator...
4, =(7,5), u =(-3,-1) 2) Let v = (1,-5), v = (-2,2) and let L be a linear operator on Rwhose matrix representation with respect to the ordered basis {u,,,) is A (3 -1 a) Determine the transition matrix (change of basis matrix) from {v, V, } to {u}. (Draw the commutative triangle). b) Find the matrix representation B, of L with respect to {v} by USING the similarity relation
2) Let 4 =(0,5), 4, =(-3, -1) v; = (1,-5), v, =(-2,2) and let L be a linear operator on R? whose matrix representation with respect to the ordered basis {u, uz} is 2 A= a) Determine the transition matrix S) (change of basis matrix) from {v, v,} to {u,,u,} (Draw the commutative triangle). I
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
With explanation! 3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
Consider the linear operator, L. on Pdefined by L(P) = p(3)x3 + p(2)x2 + P(1)ą + p0). Find the matrix representation of L with respect to the standard basis of P {1, 2, 2, 23).
12. Consider the linear operator from R² to R² defined by matrix B. (5%) and v= {f: (350-9"), both in the standand ordered basis. @ Show that vis a basis for R. Find matrix K to express the lineat operator in the basis v
1. Find a matrix A such that L(x) = A ∗ x for all x ∈ R³ .What is the relation between A and the matrix representation eLe of L with respect to the standard bases for R³and R∧4? 2. 3. Compute the matrix representative eLS of . Let L : R3 → R4 be the linear transformation given by L 22 23 [(3x1 – 2x2 – 7x3)] (5x1 – 3x3) (4x2 – 3x3) [(6x1 + 2x2 – 3x3) Let...
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d)...
2. Suppose the linear operator L:R2 + R2 has matrix representation A = (Lee = (_} -). with respect to the basis E = [(1,1), (1, -1)7]. (a) Find B = [L], with respect to the basis F= |(1,0), (2, 1)T] .