Math 1b Practical — Problem Set 4Due 3:00pm, Monday, January 31, 20111. (15 points) The idea of Theorem 4 of the handout on Bases for row and null spaces canbe extended. If M is a matrix in basic form, then it is pretty easy to write d own anotherbasic matrix whose rows form a basis for the null space of M.LetM =⎛⎝0 a 01d1 b 00e0 c 10f⎞⎠.(i) Fill in the blanks in the matrix b elow so that its rows are in the null space of M.N =• 0 ••1• 1 ••0(ii) Explain briefly why the rows of N are linearly independent and why they span thenull space of M. Don’t try to say things in general, but, for the second assertion, simplywrite down the three equations that mean (x1,...,x5)isinthenullspaceofM and showthat it is then a linear combination of the rows of N.2. (15 points) Find the area of the triangle with vertices (1, 0, 1), (2, −1, 1), and (1, 2, 3).These points are in R3. [Suggestion: You can translate the triangle so that one of itsvertices is the origin. Now we are asking for the area of a triangle with vertices 0, x, y,and this is one-half of the ar ea of the parallelogram spanned by x and y.]3. (30 points) Let v1, v2, v3be the rows ofA =⎛⎝1 1 1 1111−3 −2 −101239 4 1 0149⎞⎠.(i) Calculate the orthogonal vectors u1, u2, u3produced by the Gram-Schmidt processstarting from the vi’s and take them as the r ows of a m atrix B. [The calculations aren’thard because some of the inner products you must compute are 0.](ii) Give 3 × 3 lower triangular matrices E and F so that B = EA and A = FB.(iii) Find the orthogonal projection of x =(3, 2, 1, 0, 1, 2, 3) onto the row space U ofA. Express it as a vector, as a linear combination of the ui’s, and as a linear combinationof the vi’s.(iv) What is the reflection of x through U?4. (20 points) Let V be the vector space of continuous real-valued functions on the interval[0,π]. Define an inner pro duct ·, · on V byf,g =π0f(x)g(x) dx.Fact: The functions sin(x), sin(2x), sin(3x),... are pairwise orthogonal with respect tothis inner product, and the squared-lengthsπ0sin(kx)2dx of these functions ar e all thesame (no proof required, but check this with Mathematica).Use M athematica (see the attached page for help) or some other program to com-pute the orthogonal projection h(x) of the function 1 onto the 7-dimensional subspace Uspannned by sin(x), sin(2x),...,sin(7x). Sk etch the graph of h(x)from0toπ.Whatisthe distance from h(x)to1?5. (20 points) Evaluate the determinants of the following matrices. Do the first with theusual formula for 3 × 3 determinants, and do the second by using row operations.(i)⎛⎝21−112 3xy z⎞⎠(When the determinant is set equal to 0, we have the equation of a plane in R3.Thisis the plane spanned by the firs t two rows; we will talk about this later. Check that thefirst two rows are in the plane, but don’t turn this in.)(ii)⎛⎜⎜⎜⎝3111113111113111113111113⎞⎟⎟⎟⎠[Suggestion: Do the following row operations: Add all rows to the bottom row. Thensubtract scalar multiples of the new b ottom row from each of the top four rows so as tochange the 1’s to 0’s. At that point, you should have a lower triangular matrix.]In[22]:=Integrate Sin5x Sin13 x, x, 0, PiOut[22]=0In[23]:=IntegrateSin13 x^2, x, 0, PiOut[23]=2In[26]:=PlotSinx  2 Sin2x  Sin4x  Sin7x, x, 0, PiOut[26]=0.5 1.0 1.5 2.0 2.5