2 4 la Use the GramSchmidt process to find an orthogonal bas

2 4. la Use the Gram-Schmidt process to find an orthogonal basis for the span of 0 5 b Find an orthonormal basis for the subspace spanned by the two vectors above 1 2 2 6 3 1 2 0 0 1 2. The matrix A 1 2 4 12 5 is row-equivalent to 0 0 1 31 0 0 0 0 0 1 2 2 6 3 a) What is the rank of A? b) Find a basis for the column space of A. c Find a basis for the null space of A. d) What is the dimension of the null space of A? 3. Is 1 in the span of 2 2 Give a reason for your answer. 4s -5t s, IER 4. Find a basis for 3s-21 5a Find the standard matrix for the linear transformation T R2 R2 that rotates each vector thirty degrees clockwise. b Find the standard matrix for the linear transformation S: R2 R2 that shears by sending ei to e, and e, to 3e, e c) Find the standard matrix for the linear transformation that first shears as in b), then rotates as in a). 6. Are the matrices invertible? Give a reason for your answer. 2 1 4 5 2 3 0 1 0 5 20 40 4 5 -6 3 0 30 15 7. If A is a 4x7 matrix and the null space of A has dimension 5, what is the dimension of Col A? 8. Let B and x 3 Find Ix 7x, 5x 9. If T: R2 R3 is defined by T -2x1 give the standard matrix for T. x, 2x,

Solution

Please mention clearly which part you want to be answered

answering 15th question)

AX=B

X =[X1 X2 X3]^T

X1 + 3X3 = -1 ------------------1

2x1 +x2 +x3 = 5 ----------------2

2x1 +3x2 + x3 = 9 -----------------3

eqn 3 -eqn 2

2x2 =4 : x₂=2

in eqn 3 : 2x1 +3(2) +x3 = 9

2x1 + x3 = 3

from eqn 1 : X1 + 3X3 = -1

x1 +3(3 -2x1) = -1

x1 +9 -6x1 =-1

-5x1 =-10

x₁ = 2 ; x₃ =-1

Solution X =[2 2 -1]^T