Q3. Consider the following linear system: t+y+z=7 [; +y2z=5 -+z=3 {a) Reduce the linear systeam to upper triangular form by row operations. You should ehcountar a step that requires a row exchanga. {b) Explain why this row exchange is necessary in plain English but use a more precise language to pinpoint the entry(s) imvolved, as well as what arthmetic operation failled. Then, describe what the effect of this row exchange has to the underlying linear system by campleting this sentence: The exchange of row _____ with row _ is equivalent to renaming the 3 equationasthe ___ equationin the linear system before that step, where the linear system (as a set of simultaneous scalar equations) was: (e} Finalty, circle the pivots in the reduced upper triangular system, and then trace your row operations backwards and underline AL L the numbers involved in the arithmetic to produce the pivots in the reduced system. Q4. Consider the following linear system that is very similar to that in Q3: t+y+z=7 x+y2z=5 x=y+z=3 {a) Reduce the linear systam to upper triangular form by row operations. You should encountar a step with an unrecoverable difficulty. {b) Explain why the Gaussian Elimination (G.E.) method cannot be continued after the step you found in (a). {c) Finalty, circle the pivots in the reduced system. How does the number of pivots compare to the previous system in Q37 State why this system in Q4 has an unrecoverable difficulty in terms of its number of plvots. Q5. Consider the following Linear system that is very similar to that in Q4: t+y+z=0 t+y-z=0 - y+z=10 (a) Solve this system WITHOUT writing down the steps row operations. Instead, describe why the row operations from Q4 still apply, and state what numbers you need to change for the final reduced system. (b} Write down the general solution by introducing some free variable(s). Are there 0, 1 or oo solutions? Sketch the solutions as you would sketch {eif] B} as you did in Problem Set #1. Q6. Consider the general 2 x 2 linear system Av = w, where: A = 2 2. w = ] involves some numbers a, b, c, d, e, f with a # 0, and B = [y] is is the unknown vector. (You do not need to substitute your student ID for these 6 numbers) (a) Use ONE step of row operation with justifications to make this system upper triangular. (Note: you should specify the row operation and the result of it, and what condition you applied to perform the row operation] (b) Compute the product of the diagonal entries of the upper triangular matrix from (a). This is product is known as the determinant of the 2 x 2 linear system, we use the notation det A for the determinant of A because it depends on all the entries of A but not on w. (c) Suppose det A # 0, finish solving the linear system and write down the general solution in terms of a, b, c, d, e, f.Q7. By inspection, one can see that the point (12,0,0) is on the plane: x = 12 + 3y + z. (a) By inspection, argue that all position vectors on this plane will take the form: =+ where s and t are free parameters and the * entries are to be replaced by numbers given in the first sentence. [Hint: you could consider the set of three scalar equations of x, y,z corresponding to the vector equation above.] {Note. \"By inspection\" here means that a correct solution would not see the need of arithmetic operations or algebraic manipulations, but only substitutions. You would just need to copy the right numbers to the right places.) (b) Note that the scalar equation of the plane can be written as a matrix-vector equation form: (c X [i: =3 1] [y] =[12] Solve by writing down the standard procedure of G.E., verify that the linear combination form of the solution vector you gave in part (a) with two free parameters s, t, is indeed the general solution to the \"linear system\" x = 12 + 3y + z. [Hint: in this \"system\" of unusual size, you will find that part of the standard G.E. process becomes \"nothing to be done\". More straightforwardly, the work here is not about figuring out row operations but identifying pivot(s) to explain why you can proceed to substitution step, leading to the two free parameters - this also is part of the G.E. process, and not just the row operations]. Plugin the three vectors below into the general solution (with the * values replaced by the correct values): * * * e} o4 tpd 1 into the matrix-vector form equation in (b). Some of them won't solve the equation. What is their geometrical relationship with the rowvector [1 -3 1]