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Part 1: Calculate the dimension Given the following matrix: 1 1 A 4 3 -2 4 5 -2 16 9-9 -4 4 3 4 -1 Let S be the space spanned by A, that is S span(A) The dimension of Sis: dim S = 1 Part 2: Find a basis Enter a basis B of S below. 0 | -3 7 B=BA 0 0 -1 1 0 0 0 0
4. (8 marks) Let V be the vector space of solutions to the ODE y" hyperbolic functions y 0, spanned by the cosh r and y2 = sinh r, and let z1 = e and z2 = e = (a) Show that 21, %2} is a basis for V {1, 2to {yı, Y2}. Show all working (b) Find the transition matrix from the basis 3 4. (8 marks) Let V be the vector space of solutions to the ODE y"...
1 4 2 1 7.[12pts) Let A = 0 1 1-2 -8 -4 -2 (a) Find bases for the four fundamental subspaces of the matrix A. Basis for n(A) = nullspace of A: Basis for N(4")= nullspace of A": Basis for col(A) = column space of A: Basis for col(A) = column space of A': () Give a vector space that is isomorphic to col (A) N(A).
for the question, thanks for your help! 2. Let 2 -2 -11 1 3 S1 8 and b -2 -5 7 A= -4 5 2-9 18 Moreover, let A be the 4 x 3 matrix consisting of columns in S (a) (2.5 pt) Find an orthonormal basis for span(S). Also find the projection of b onto span(S) (b) (1.5 pt) Find the QR-decomposition of A. (c) (1 pt) Find the least square solution & such that |A - bl2 is...
Consider the matrix 0 4 8 24 0-3-6 3 18 A-0 24 2 -12 0 -2-3 0 7 0 3 5 [51 [51 a) Find a basis for the row space Row(A) of A (b) Find a basis for the column space Col(A) of A (c) Find a basis space d) Find the rank Rank(A) and the nullity of A (e) Determine if the vector v (1,4,-2,5,2) belongs to the null space of A. - As always,[5 is for the...
1. 2. 3. 4. 5. Given that B = {[1 7 3], [ – 2 –7 – 3), [6 23 10]} is a basis of R' and C = {[1 0 0], [-4 1 -2], [-2 1 - 1]} is another basis for R! find the transition matrix that converts coordinates with respect to base B to coordinates with respect to base C. Preview Find a single matrix for the transformation that is equivalent to doing the following four transformations...
_Determine which of these sets is a basis for R3. 1 - 1 ^ {[:] 7] [8]} - {[7] | c{[7] [8] [8]} » {[![10]} Determine the dimension of the subspace W = span {V1, V2, V3, V4} where A. dim(W)=1 B. dim(W) = 2 C. dim(W) = 3 D. dim(W) = 4 8 Determine both the rank and nullity of the matrix A= [1 0 1 | 2 -3 4 -2 -2 2 -4 1 0 3 -4 2...
Let V = M2(R), and let U be the span of S = 2. (a) Let V = M,(R), and let U be the span of s={(1 1) ($ 3). (3), (1 9). (1) 2.)} Find a basis for U contained in S. (b) Let W be the subspace of P spanned by T = {2} + 22 – 1, -2.3 + 2x +1,23 +22² + 2x – 1, 2x3 + x2 +1 -2, 4.23 + 2x2 - -4}. Find...
1. Let 1 -1][-1 s={ 112 [1] 1 1 Find a basis for the subspace W = span S of M22. What is the dim W? 2. Find the basis for the solution space of the homogeneous system: a. x+2y = 0 2x+4y =0 b. 3x+2y+4z=0 2x+ y - Z = 0 x +y +3z =0
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...