Given, and
Let u = , v = , p = , q = .
(e) Now projection of X = [x,y,z]t on W using the basis B1 is :
i.e.,
i.e.,
i.e.,
Now,
Therefore, R = .
(f) Now projection of X = [x,y,z]t on W using the basis B2 is :
i.e.,
i.e.,
i.e.,
Now,
Therefore, R = .
(c) Now, R2 = R*R
i.e., R2 =
i.e., R2 = [Using Calculator]
i.e., R2 = R
Geometric meaning of this equality is :
If we apply this transformation twice on a vector, then it will return the effect of the first transformation.
6.20 Below, you are given two sets of vectors B, and B, in R. B =...
1) for R2 Given the vectors b1,b2, C1, and cz. B = {b1,b2} is a basis for R2C = {C1,C2} is a basis b = [i.bz = [33],4 = (-2) c2 = [4] (a) Find the change of coordinates matrix to convert from B to C. (b) Find the coordinate vectors [x]B, [x]c, lyle and [ylc given x = [11] y = [12]
Consider the linear transformation T: "R" whose matrix A relative to the standard basis is given. A=[1:2] (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (11, 12) = 2,3 |_) (b) Find a basis for each of the corresponding eigenspaces. B = X B2 = = {I (c) Find the matrix A' for T relative to the basis B', where B'is made up of the basis vectors found in part (b). A=
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansion x = 2e1 + e2. Find expansion of the vector x in another basis 1 -2 er, e2, of X, if the change of basis matrix from the basis e to the basis e, s (b) In a two-dimensional linear space X vectors el, e2 forn a basis. In this basis a vector r E...
(1 point) Consider the basis B of R consisting of the vectors and Note: These vectors are written in terms of the standard basis, E. You know the following about e R2: - [ 6 B Find [피e. TE (1 point) Consider the basis B of R consisting of the vectors and Note: These vectors are written in terms of the standard basis, E. You know the following about e R2: - [ 6 B Find [피e. TE
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)...
Let B= {[3]• [4]) and c = ( ) [1]} be two basis for R. (1) Suppose Find x, y, are these vectors equal? What does this mean geometrically? i.e. draw x and y in a plane as a linear combination of vectors in B and C. (2) Let u Find the corresponding coordinate vectors us and ſuc. What does this mean geometrically? (3) Find the change of coordinate matrix Pg and use Pg to compute us from part (2).
Question 2. [2] Given vectors: a= Z = b = у = 2 3 | 4 3 3 3 2 3 3 3 3 let A be a 3 x 4 matrix such that projw z =a where W = Col(ATA) [1] (a) Find (and explain) a non-trivial solution for Ax = 0 (b) Explain whether or not projy y = b where V = Col(AAT) [1]
In R. let V be the orthogonal complement of the vectors u and v, where u = (1,9, 3,61) and v= (4, 36, 13, 254) Find a basis B = {b1,b2} for V: b = 1 Now find five vectors in V such that no two of them are parallel e- LLL
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
Assume that the transition matrix from basis B = {b1, b2, b3} to basis C = {c1, c2, c3} is PC,B = 1/2*[ 0 -1 1 ; -1 1 1 ; 1 0 0 ]. (a) If u = b1 + b2 + 2b3, find [u]C. (b) Calculate PB,C. (c) Suppose that c1 = (1, 2, 3), c2 = (1, 2, 0), c3 = (1, 0, 0) and let S be the standard basis for R 3 . (i) Find...