5. Is the following sequence of vectors a basis for RX ? Verify that it is,...
#8
B Lets-(0+4 r+e34 r3 ) is aha is ofs. +aite 2+3a3 =0]. Verify that iS a basis of S In exercises 8 12 decide if the sequence B is a basis for the space S of exercise 7 See Method (5.3.2). Note that if there are the right number of vectors you still have to show that the vectors belong to the subspace S
B Lets-(0+4 r+e34 r3 ) is aha is ofs. +aite 2+3a3 =0]. Verify that iS...
Exercise 11. Given th=3(1 1 1-1)" and v2-(-1 1 3 5)T, verify that these vectors form an orthonormal set in R. Extend this set to an orthonormal basis for R4 by finding an orthonormal basis for the nullspace of 1 -1 113 5 Hint: First find a basis for the null space and then use the G-S process.
Exercise 11. Given th=3(1 1 1-1)" and v2-(-1 1 3 5)T, verify that these vectors form an orthonormal set in R. Extend...
4 Q1. Consider the following set of vectors3,0 4 (a) Show that these vectors are linearly independent. (b) Do these vectors span a plane? Explain your answer. (c) Is the set a basis for R5? Why, or why not?
4 Q1. Consider the following set of vectors3,0 4 (a) Show that these vectors are linearly independent. (b) Do these vectors span a plane? Explain your answer. (c) Is the set a basis for R5? Why, or why not?
2. Consider the following sequence x(n) = cos (2n/3)sin(2ton/5). a. Is the sequence periodic? If yes, what is the period? If no, why not? Note: you need to show your work analytically but you can verify that your answer is correct by sketching the sequence using Matlab b. Find and sketch the complex exponential Fourier series coefficients (Magnitude and Phase). Verify using Matlab. Include code and graphics.
4. The following vectors form a basis for R. Use these vectors in the Gram-Schmidt process to construct an orthonormal basis for R'. u =(3, 2, 0); uz =(1,5, -1); uz =(5,-1,2) 5. Determine the kernel and range of each of the following transformations. Show that dim ker(7) + dim range(T) = dim domain(T) for each transformation. a). T(x, y, z) = (x + y, z) of R R? b). 7(x, y, z) = (3x,x - y, y) of R...
Question 15: Do the vectors below form a basis for R3? If so, explain. If not, remove as many vectors as you need to form a basis and show that the resulting set of vectors form a basis for R3. -- () -- () -- ().- 0 1
a) Verify that B is a basis for IR3 (b) Use the Gram-Schmidt process to produce an orthogonal basis for R (c) Normalize the vectors to produce an orthonormal basis for R3.
Suppose that u and v are non-zero vectors in Rn. Verify that the two vectors u and v - (u.v/u.u)u are orthogonal. Then pick two specific vectors u and v in R2. Plot the three vectors, u, v and v - (u.v/u.u)u on the same graph. Explain the geometric significance of v - (u.v/u.u)u
(1 point) Perform the Gram-Schmidt process on the following sequence of vectors. 4 -3 5 8 0 -5 X= y = ,Z = 8 -3 -2
Linear Algebra:
1. 1.9 #6 For the following W = Span({(2,6,5,-4),(5,-2,7,1),(3,-8,2,6)}) a. Assemble the vectors into the rows of a matrix A, and find the rref R of A. b. Use R to find a basis for each subspace W, and find a basis for W as well. Both bases should consist of vectors with integer entries. c. State the dimensions of W and W and verify that the Dimension Theorem is true for the subspaces.