Problem 3 Let L:R4 + R3 be given by L - (C)- [. (3x1 – 422 + 11x4) (1522 + 9x3 – 2124) a) [4 pts] Show that L is a linear transformation, and find the matrix representation A of L with respect to the standard bases for R4 and R3. b) [3 pts] Use part a) to find a basis for ker(L). c) [3 pts] Use part a) for find a basis for im(L).
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))
Problem 24 : Let b b2 b3 ba E R4 be a fixed vector, b + 0. Define L:R4 R by C1 12 L(x) = b. I, 2= ER4. 13 24 where bºx is the dot product of b and x in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L...
Tbi b2 Problem 24 : Let b e R4 be a fixed vector, b+0. b3 b4 Define L:R4 → R by 11 12 L(x) = 6-2, x= ER 23 24 where b.x is the dot product of b and 2 in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L onto?...
Let L : R2 → R3 be a linear transformation such that L 1 1 = 1 2
3 and L 1 2 = 2 1 3 . Find L 2 1 Find the standard matrix
representing L. Find the dimensions of the kernel and the range of
L and their bases.
12. Let L : R² + RP be a linear transformation such that L | (3) - -(5)-(1) Find I (*) Find the standard matrix representing L. Find...
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...
Let L: R3 --> R3 be defined by
Only need c-e solved.
6, (24 points) Let L : R3 → R3 be defined by (a) Find A, the standard matrix representation of f (b) Let 0 -2 2. Check that倔,G, u) is a basis of R3. (c) Find the transition matrix B from the ordered basis U (t, iz, a) to the standard basis {e, е,6). For questions (d) and (e), you can write your answer in terms of A...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
[-16/257 128. In R3, ū1 12/25 form an orthonormal basis of V = span(ün, ). 3/ 51 (A) Find the 3 x 3 matrix P such that projv (T) = Pi for all i ER (B) Find the 3 x 3 matrix P, such that projv) = P27 for all E R. (C) Use either V or V - to fill in the blanks: ker(projv) = _ im(projv) = _ ker(projy) = __ im(projy) = _
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