suppose that T:R^3 →R^2 is such that T(e1)= [ 2] T(e2)= [ 1 ] T(e3)=[ 0 ] [ 1 ] [ 1 ] [ 1] and suppose that S : R^2 → R ^2 is given by the projection onto the x axis (a) What is the matrix S◦ T? (b) What is the kernel of S◦T?
11.) Let T:R" - R"be a linear transformation. Prove T is onto if and only if T is one-to-one. 12.) Let T:R" - R" and S:R" - R" be linear transformations such that TSX=X for all x ER". Find an example such that ST(x))+x for some xER". - .-.n that tidul,
(1 point) If T:R → R is a linear transformation such that 13 , T||0||= 01) [ 1] T||1||= -1, Uo4 -4 i 2 1 T||0||= (11) then T|| -2
1. (a) Let T:R' R'be defined by T(x) = 5 -2. Is T a linear transformation? If so, prove that it is. If not, explain why not. (b) More generally than part (a), suppose that T:R → R is defined by T(x) = ax +b, where a and b are constants. What must be true about a and b in order for T to be a linear transformation? Explain your answer.
A Linear transformation T:R^5→R^4 is given as
How do I find the standard matrix of T, the zero space and
column-space of T?
How do I find the rank and the dimension of the zero-space of
T?
C1 x2 1 as C2 + 4- x5 C4 C5
Consider the matrix transformation T:R → R given by T(x,y,z) = (x+ay, x+(a+1)y, x+ay+z) where a = 13. First use inverse of transformation to find T-(2,1,2). if T-(2,1,2)=(b,c,d) then b+c+d =
T:R R2 is a linear transformation with T(1,0, 2) = (2, 1) and T(0,1,-1) = (-5,2). It follows that T(2, -3,7) is equal to Select one: 0 a. (-11, -8) O b. (2, 3) c (19, -1) d. not enough information is given to determine the answer e(-3,3)
Let T:R" → Rh be the ödentity map T(*)=² show that undor any coordmate system the matrix form of Tis - In are isomosphic 2) which of the following to UR3? UR <R ii) {(2,4,0w) (x,ywelf? iv) IR
Question 1 (12 points) Determine the following linear maps of vector spaces over R are isomorphism or not. If it is an isomorphism, find its inverse map. (Hint: inverse of matrices.) If it is not an isomorphism, briefly explain why (1) (Rotation by 60o) a 3 V31 (2) (Reflection about z-axis)
Given real numbers a and b, find a linear transformation T:R^3→R^3 such that the range of T is the plane z=ax+by.