Find the Kernel and the Range (Image) for the operators (1 1 2 -2 2 2
1. Find a basis for the image and kernel of the following matrices. It does not have to be an orthonormal basis, but don't include more vectors than needed. (1 1 1\ (a) A= (1 2 3 11 3 5) 12 2 21 (b) A = 2 2 2 12 2 2) 11 11 (c) A= 1 3 11 -1) NNN
3. Find the Kernel and the Range for the projection operator onto the plane z = 0 in the basis 7,J,K
3. Find the Kernel and the Range for the projection operator onto the plane z = 0 in the basis 7,J,K
a. find the form of Kernel(T)
b. find the form of range(T)
P2 be the mapping defined by 23. Let T: P -
P2 be the mapping defined by 23. Let T: P -
(2 points) Let 4- -1 01 1 1-1 0-2]. Find orthonormal bases of the kernel, row space, and image (column space) of A (b) Basis of the row space: (c) Basis of the image (column space)
Problem 2. Describe the kernel and image of the transformation that is a rotation throngh an angle of in the counterclockwise direction.
Let T(f(t)) = t(f(t)) from P to P. Find the image and kernel of T.
5. Characterize the vectors (X.X.2) in the range T (R) and those in the kernel ker(T) in terms of concrete relations among the coordinates xyz for the linear transformation T: (847) ER3 7—(x - y + 22, 2x + y -x - 2y + 2x) ER3. What are the dimensions of the range and the kernel of T?
In the following transformations:
a)Find the Kernel and Image
b)Find dimK(T) and dimI(T) and show that
dimK(T)+dimi(T)=dimV
c)say if the transformations are injective, suprajective or
bijective
i)
such that:
ii),
such that:
iii)
, such that:
T:R? → R Tx,y,z) = (x - y, 22) T: M2:3 (R) → M2.2(R) SZD LED TEDT SID ZID IIDL sip-sip tip-tip) = T: P2(R) + P(R) T(p2)) = rp() +p (2)
Finding the Nullity and Describing the Kernel and Range In Exercises 33–40, let T: R3→R3 be a linear transformation. Find the nullity of T and give a geometric description of the kernel and range of T. T is the reflection through the yz-coordinate plane: T(x, y, z) = (−x, y, z)
Find a 2x2 matrix whose 2-eigenspace is the line through (2,-1) and whose kernel is the line through (1, 2). Graph is shown below. kernel 2-eigenspace