11.) Let T:R" - R"be a linear transformation. Prove T is onto if and only if...
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.
Let A= and 6 = Define the linear transformation T:R? +R by T'(X) = Ai. Find a vector # whose image under T' is 6. Is the vector i unique choose choose unique Submit answer not unique
13. (Linear Algebra Review) Recall that a function T:R" + R" is one-to-one if for each b ER", there is at most one x ER" such that T(x) = b. Let T be a linear transformation given by a matrix A. If T is one-to-one, then what can be said about the columns of A? What can be said about the determinant of A?
(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
Q22 A` = AP, B` = BQ 5.4 Composition of Linear Transformations229 Let T be the linear transformation from P3 over R to R2x2 defined by ao T (ao+ ax azx a3x) ao t a3 a3 Find bases A' of Pa and B' of R22 that satisfy the conditions given in Theorem 5.19. 23. Let T be the linear transformation from R2x2 to P2 0ver R defined by a12 a22 +(a1-a22)x +(a12 -a21)x T a22 Find bases A' of R2x2...
How was the linear transformation of b1 and b2 were applied (L(b1) , L(b2))? NOTE: b1=(1,1)^T , b2=(-1,1)^T Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R? into itself and defined by where (bi, b2] is the ordered basis defined in Example 3. Find the matrix A represent- ing L with respect to [bi, b2l Solution Thus, A0 2 onofosmation D defined by D(n n' maps P into P, Given the ordered Linear Transformations EXAMPLE 4 Let...
Let T be the linear transformation of R* to R that takes and to and takes and - a) Explain why there is one and only one linear transfomation from R to R that does this. b) Find bases for both the kernel and the range of this linear transformation
Question 5. (20 pts) Let T: R² + R* be a linear transformation such that T(21, 12) = (x1 - 2x2, -21 +3.22, 3.21 - 202). (1). Find the standard matrix of T (call it A). (2). Is T one-to-one? Justify your answer. (3). Is T onto? Justify your answer.
true or false The linear transformation T:R? R? defined by T'(x,y) = (x + y,X-V) is invertible.
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).