7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
11. Let Vi,b, . . . , vn, and u be vectors in Rm. Determine whether or not the following two statements mean the same thing. Explain your answer. Statement l: u is in Sp(vi, V2,. . . , Vn). Statement 2: If A-Vi U2 Vis an m x n matrix, there exists a vector i in R" such that A
nsid r the following et ār vnctors. Let 1 v2 and V3 be column vectors in and let A be the 3 × 3 matrix v 1 v2 v③ with these vectors as its columns. The vi v2 and ] are linearly dependent if and nly the hom 9ene us linear system with augmented matrix 시 has a no tr ia solution Consider the following equation. 81-3-311 Solve for ci 2, andc3. If a nontrlvial solution exists, state it or...
Let V be a vector space. Suppose dimV = n and {V1, V2, ..., Vn} is a basis of V. Thei which of the following is always true? a) Any set of n vectors is linearly dependent b) Any linearly dependent set in V is not part of basis c) Any linearly dependent set with n - 1 vectors is a basis d) A linearly independent set with n vectors is a basis
2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I,
2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is orthogonal, that is PT P I,
Can I get help with questions 2,3,4,6?
be the (2) Determine if the following sequences of vectors vi, V2, V3 are linearly de- pendent or linearly independent (a) ces of V 0 0 V1= V2 = V3 = w. It (b) contains @0 (S) V1= Vo= Va (c) inations (CE) n m. -2 VI = V2= V3 (3) Consider the vectors 6) () Vo = V3 = in R2. Compute scalars ,2, E3 not all 0 such that I1V1+2V2 +r3V3...
Let u = and v= Determine whether the vectors u and v are linearly independent or linearly dependent, and choose the most correct answer below. A. The vectors are linearly independent. B. We cannot easily tell whether the vectors are linearly independent or linearly dependent. C. The vectors are linearly dependent.
Consider the three 4-dimensional vectors aj = _21, 22 = 1 , a3 = 11 and the matrix A = [a], 22, az). (a) Find rank A and null A. (b) The linear transformation TA : R3 → R4 is defined by T.(x) = Ax. Determine whether TA is injective or not. (c) Determine whether the vectors aj, a2, az are linearly independent or dependent.
Please explain why each is
true or false (examples would be helpful thank you!)
2. Let vi,.. . ,vn E R", and let A - [vi. .. vn]. Suppose A-b has no solutions for some b. Circle all true statements b) (vi,..., Vn) is linearly independent c) Span(vi,... , vn) f R" d) Span(v1, . . . ,%) = Rk where k 〈 m e) Span(v1, . . . , vn) has dimension k where k < m f) Ax0m...
Question 1: Let T: R3 ---> R2 defined by T(x1,x2,x3) = (x1 + 2x2, 2x1 - x2). Show that T as defined above is a Liner Transformation. Question 2: Determine whether the given set of vectors is a basis for S = {(1,2,1) , (3,-1,2),(1,1,-1)} R3 Need answers to both questions.