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- Suppose A is equal to the following: A= 1 3 2 -1 2 -6 2 2 a) Determine N(A), the nullspace of A.b) Show that the vectors in your answer in part (a) span N(A) and are linearly independent.c) Do the vectors in your answer to part (a) span R4?arrow_forwardi) Solve the following system of equations using Gauss (or Gauss-Jordan) elimination: 3x+y =-2 4x +3y=-1 -2x+y =3 ii) Based ONLY on your answer in part (i) and without doing any extra work, answer True or False to the following statements and briefly explain why they are True or False (no explanation: no marks). 1) Vector w = (-2, -1, 3) belongs to the spanning set of vectors u = (3, 4, -2) and v = (1, 3, 1). 2) The set S = {(-2, -1, 3), (3, 4, -2), (1, 3, 1)} is a basis for its spanning set. 3) The following system of equations has a unique solution: 3x +y -2z=-1 4x +3y-z =0 -2x+y +3z=4arrow_forwardIf kk is a real number, then the vectors (1,k),(k,k+2) are linearly independent precisely when k≠a,b,where a = _____ , b = ________________ , and a<b.arrow_forward
- Questions (a) and (b) in this problem refer to the vectors u = 1 −2 2 and y = −2 1 −1(a) Find the orthogonal projection of y onto the line through u.(b) Write y as the sum of two vectors, where the first is in Span{u} and the second is orthogonalto u.arrow_forward(a) Suppose that a and b are two unit vectors in R^3, and a·b=11/24. Find the norm of the vector 4a+ 3b.arrow_forwardThe zero vector 0 = (0,0,0) can be written as a linear combination of the vectors v1, v2, and v3 because 0= 0v1+0v2+0v3. This is called the trivial solution can you find a nontrivial way of writing as a linear combination of the three factors?(enter your answer in terms of v1, v2, v3. if not possible, enter impossible)arrow_forward
- The zero vector 0=(0, 0, 0) can be written as a linear combination of the vectors v1, v2, and v3 because 0=0v1+0v2+0v3. This is call the trivial solution. Can you find a nontrivial way of writing 0 as a linear combination of the htree vectors? (Enter your answer in terms of v1, v2, and v3. If not possible, enter IMPOSSIBLE.)arrow_forwardFind 2?, −3?, ? + ?, and 3? − 4? for the given vectors ? and ?. u=(-2,4) and v=(4,-2) 2u= -3v= u+v= 3u-4v=arrow_forwardIn Problems 23–25, use the vectors u = 2i - 3j + k andv = - i + 3j + 2k.23. Find u * v.24. Find the direction angles for u.25. Find the area of the parallelogram that has u and v asadjacent sides.arrow_forward
- Given v=[2,-9] and w = [7,-8], what is 2w-v written as a linear combination of unit vectors? please show how you came up with thisarrow_forwardLet w,x,y,z be vectors and suppose z=−4x−3yand w=−16x+4y−4z Mark the statements below that must be true.arrow_forward6) a. Suppose the columns of A are linearly independent. Given any b, what are the possible numbers of solutions for Ax = b? Justify your answer.b. Suppose the columns of A are linearly dependent. Given any b, what are the possible numbers of solutions for Ax = b? Justify your answer.7) Give an example of 3 nonzero vectors v1,v2,v3 ∈ R3 such that v3 is not a linear combination of v1 and v2, but {v1,v2,v3} is a linearly dependent set. Justify your answer.8) Show that {v1,v2,v3,v4} is linearly dependent if {v1,v2,v3} is linearly dependent. (In general, this proves that a set is linearly dependent if any subset is linearly dependent.)arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning