3. (i). Let A =
2 |
1 |
4 |
1 |
2 |
5 |
The RREF of A is
1 |
0 |
1 |
0 |
1 |
2 |
This implies that [x]E = (1,2)T.
Also, let B =
4 |
1 |
4 |
3 |
1 |
5 |
The RREF of A is
1 |
0 |
-1 |
0 |
1 |
8 |
This implies that [x]F = (-1,8)T.
(ii). Let M =
4 |
1 |
2 |
1 |
3 |
1 |
1 |
2 |
The RREF of M is
1 |
0 |
1 |
-1 |
0 |
1 |
-2 |
5 |
Hence the transition matrix S from the basis E to the basis F is
1 |
-1 |
-2 |
5 |
(iii). S[x]E = (-1,8)T= [x]F.
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
Exercise 1: (20pts) Let u1-11, 1, 1)T, u2-(1, 2, 2)T, u,-(2, 3, 4)T, ν,-(4,6,7)T, v2 = (0, 1,1)1 , V3 = ( ) (a) Find the transition matrix from fvi, v2, vs] to sui, u2, us] (b) If x 2vı +3v2 - 4vs, determine the coordinates of x with respect to fui, u2, us] 0,1,2
UCICI 26. Let S = {V1, V2} and T = {w1, W2} be ordered bases for R?, where 38. If the transition matrix from S to T is determine T.
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
Problem 4 A set of vectors is given by S = {V1, V2, V3} in R3 where eV1 = 1 5 -4 7 eV2 = 3 . eV3 = 11 -6 10 a) [3 pts) Show that S is a basis for R3. b) (4 pts] Using the above coordinate vectors, find the base transition matrix eTs from the basis S to the standard basis e. Then compute the base transition matrix sTe from the standard basis e to the...
Please do only e and f and show work
null(AT) null(A) T col(A) row(A) Figure 5.6 The four fundamental subspaces (f) Find bases for the four fundamental subspaces of 1 1 1 6 -1 0 1 -1 2 A= -2 3 1 -2 1 4 1 6 1 3 8. Given a subspace W of R", define the orthogonal complement of W to be W vE R u v 0 for every u E W (a) Let W span(e, e2)...
[1] (a) Verify that vectors ul 2 | ,u2 -1 . из 0 | are pairwise orthogonal (b) Prove that ũi,u2Ф are linearly independent and hence form a basis of R3. (c) Let PRR3 be the orthogonal projection onto Spansüi, us]. Find bases for the image and kernel of P, without using the matrix of P. Find the rank and nullity (d) Find Pul, Риг, and Риз in a snap. Find the matrix of P with respect to the basis...
15 points) Consider the following vectors in R3 0 0 2 V1 = 1 ; V2 = 3 ; V3 = 1] ; V4 = -1;V5 = 4 1 2 3 = a) Are V1, V2, V3, V4, V5 linearly independent? Explain. b) Let H (V1, V2, V3, V4, V5) be a 3 x 5 matrix, find (i) a basis of N(H) (ii) a basis of R(H) (iii) a basis of C(H) (iv) the rank of H (v) the nullity...
conservation of momentum before v1=2 m1=100g v2=-3 m2= 300 find after u1=-2 u2=___?
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers dı, d2, d3 € R. (i) Show that there is a matrix A E M3(R), and only one, with eigenvalues dı, d2, d3 and corresponding eigenvectors V1, V2, V3. (ii) Show that if {V1, V2, V3} is an orthonormal set of vectors. then the matrix A is symmetric.