5. ts) Let T(x) be a linear transformation. Suppose the following ma- trix is a row echelon form of the matrix of the transformation. Use this row echelon form to answer the following questions. [17 3 -1 1 8. 1 1 -3 3 11 -1 3 16 4 (a) The inputs x are from R". What is n. (b) The outputs T(x) are from R". What is m. (c) Is T one to one? Justify your answer. (d) Is T onto? Justify your answer.

5. ts) Let T(x) be a linear transformation. Suppose the following ma- trix is a row echelon form of the matrix of the transformation. Use this row echelon form to answer the following questions. [17 3 -1 1 8. 1 1 -3 3 11 -1 3 16 4 (a) The inputs x are from R". What is n. (b) The outputs T(x) are from R". What is m. (c) Is T one to one? Justify your answer. (d) Is T onto? Justify your answer.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 1EQ

See similar textbooks

Similar questions

Let T be a linear transformation from M2,2 into M2,2 such that T([1000])=[1102], T([0100])=[0211], T([0010])=[1201],T([0001])=[3110]. Find T([1314]).
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet T=[1101]. Calculate the matrix product TD, and draw the image represented by this product. What is the effect of the transformation T? cExpress T as a product of a shear matrix and a reflection matrix. See Problem 2. 2. Verify that multiplication by the given matrix has the indicated effect when applied to the gray square in the table. Use c=3 in the expansion matrix and c=1 in the shear matrix. T1=[1001] Reflection in yaxis T2=[100c] Expansion or contraction in ydirection T3=[10c1] Shear in ydirection
In Exercises 1-12, determine whether T is a linear transformation. 8. defined by
In Exercises 1-12, determine whether T is a linear transformation. T:M22M22 defined by T[wxyz]=[1wzxy1]
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
The figure shows three outline versions of the letter F. The second one is obtained from the first by shrinking horizontally by a factor of 0.75, and the third is obtained from the first by shearing horizontally by a factor of 0.25. a Find a data matrix D for the first letter F. b Find a transformation matrix T that transforms the first F into the second. Calculate TD, and verify that this is a data matrix for the second F. c Find the transformation matrix S that transforms the first F into the third. Calculate SD, and verify that this is a data matrix for the third F.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.