Problem 1E: Consider the subspaceim(A) of 2 , where A=[2436] . Find a basis of ker(AT) , and draw a sketch... Problem 2E: Consider the subspace im(A) of 3 , where A=[111213] . Find a basis of ker(AT) , and draw a sketch... Problem 3E: Considerasubspace V of n . Let v1,...,vp be a basis of V and w1,...,wq a basis of V . Is v1,...,vp ,... Problem 4E: Let A bean nm matrix. Is the formula ker(A)=im(AT) necessarily true? Explain. Problem 5E: Let V be the solution space of the linear system |x1+x2+x3+x4=0x1+2x2+5x3+4x4=0| Find a basis of V . Problem 6E: If A is an nm matrix, is the formula im(A)=im(AAT) necessarily true? Explain. Problem 7E: Consider a symmetric nn matrix A. What is the relationship between im(A) and ker(A)? Problem 8E: Consider a linear transformation L(x)=Ax from n to m . with ker(L)={0} . Thepseudoinverse L+ of Lis... Problem 9E: Consider the linear system Ax=b , where A=[1326] and b=[1020] a. Draw a sketch showing the following... Problem 10E: Consider a consistent system Ax=b . a. Show that this system has a solution x0 in (kerA) .Hint: An... Problem 11E: Consider a linear transformation L(x)=Ax from n to m , with rank(A)=m . The pseudoinverseL+ of L is... Problem 12E: Using Exercise 10 as a guide, define theterm minimal least-squares solution of a linear system.... Problem 13E: Consider a linear transformation L(x)=Ax from n to m . The pseudoinverseL+ of L is the... Problem 14E: In the accompanying figure, we show the kernel and theimage of a linear transformation L from 2 to 2... Problem 15E: Consider an mn matrix A with ker(A)={0} . Showthat thereexists an nm , matrix B such that BA=In... Problem 16E: Use the formula (imA)=ker(AT) to prove theequation rank(A)=ker(AT) . Problem 17E: Does the equation rank(A)=rank(ATA) hold for all nm matrices A? Explain. Problem 18E: Does the equation rank(ATA)=rank(AAT) hold for all nm matrices A? Explain. Hint: Exercise 17 is... Problem 19E: Find the least-squares solution x* of the system Ax=b , where A=[100100] and b=[111] . Use paper and... Problem 20E: By using paper and pencil, find the least-squares solution x* of the system Ax=b , where A=[111001]... Problem 21E: Find the least-squares solution x* of the system Ax=b , where A=[6938210] and b=[0490] Determine the... Problem 22E: Find the least-squares solution x* of the system Ax=b , where A=[325345] and b=[592] Determine the... Problem 23E: Find the least-squares solution x* of the system Ax=b , where A=[112815] and b=[123] Explain. Problem 24E: Find the least-squares solution x* of the system Ax=b , where A=[123] and b=[327] . Draw a sketch... Problem 25E: Find the least-squares solution x* of the system Ax=b , where A=[1326] and b=[50] . Use only paper... Problem 26E: Find the least-squares solution x* of the system Ax=b , where A=[123456789] and b=[100] . Problem 27E: Consider an inconsistent linear system Ax=b , whereA is a 32 matrix. We are told that the... Problem 28E: Consider an orthonormal basis u1,u2,...,un , in n .Find the least-squares solution(s) of the system... Problem 29E: Find theleast-squares solution of the system Ax=b , where A=[11 10 1000 10 10] and b=[110101010] .... Problem 30E: Fit a linear function of the form f(t)=c0+c1t tothe data points (0,0),(0,1),(1,1) , using least... Problem 31E: Fit a linear function of the form f(t)=c0+c1t to the data points (0,3),(1,3),(1,6) , using least... Problem 32E: Fit a quadratic polynomial to the data points (0,27),(1,0),(2,0),(3,0) , using least squares. Sketch... Problem 33E: Find the trigonometric function of the form f(t)=c0+c1sin(t)+c2cos(t) that best fits the data points... Problem 34E: Find the function of the form f(t)=c0+c1sin(t)+c2cos(t)+c3sin(2t)+c4cos(2t) that best fits the data... Problem 35E: Suppose you wish to fit a function of the form f(t)=c+psin(t)+qcos(t) to a given continuous function... Problem 36E: Let S (t) be the number of daylight hours on the t th day of the year 2012 in Rome, Italy. We are... Problem 37E Problem 38E: In the accompanying table, we list the height h, the gender g, and the weight w of some young... Problem 39E: In the accompanying table, we list the estimated number g of genes and the estimated number z of... Problem 40E Problem 41E Problem 42E format_list_bulleted