Preparation problems for the discussion sections on October 14th and 16th1. Let v =11. Find the length of v. Find a vector u in the direction of v that has length 1 .Find a vector w that is orthogonal to v.2. Let u1=1√211, u2=1√21−1, and v =23. Find real numbers c1, c2such thatv = c1u1+ c2u2.3. Let V = {abcd: a + b + c + d = 0} be a subspace of R4.(a) Find a basis for V .(b) Find a vector that is orthogonal to V .(c) Can you find two linearly independent vectors that are orthogonal to V ?4. Let A =1 2 14 8 21 2 5.(a) Find an echelon form U of A. What are the column spaces Col (A), Col (U )? Are theyequal?(b) Find a basis for Col(U) and a basis for Col(A).(c) What are the row spaces Col(AT), and Col(UT). Are they equal?(d) Find a basis for the row space of A, Col(AT).5. Let B =1 1 0 11 0 0 1.(a) Find a basis for N ul(B).(b) Find two linear independent vectors that are orthogonal to Nul(B).(c) Is there a non-zero vector in R2orthogonal to Col(B)?6. Let B := {100,010,111} be a basis of R3. Let T : R3→ R3be the linear transformationthat mapsxyzin R3tozxy. Determine the matrix corresponding to T with respect to thebases B and B.17. Let I : P3→ P4be the linear transformation that maps p(t) totp(t) + p0(t)Consider the basis B = {1, t, t2, t3} of P3and the basis C = {1, t, t2, t3, t4} of P4. Determinethe matrix which represents I with respect to the bases B and C.8. True or False? Justify your answers.(a) The map T : R2→ R given by Tab=√a2+ b2is a linear transformation.(b) The map T : R2→ R2given by Tab=−bais a linear transformation.(c) If u and v in R2are such that u.v = 0 (u and v are orthogonal) then u and v areperpendicular (geometrically) to each other.(d) Let V be a subspace and u, v be two vectors in V , then v −u.vu.uu is orthogonal to u.(e) Let T : V → W be a linear transformation and v1, v2, ..., vnbe vectors in V . IfT (v1), T (v2), ..., T (vn) are linearly independent then v1, v2, ..., vnare also linearly in-dependent.(f) Let T : V → W be a linear transformation and v1, v2, ..., vnbe vectors in V . Ifv1, v2, ..., vnare linearly independent then T (v1), T (v2), ..., T (vn) are also linearly in-dependent.(g) Let T : R2→ R3be a linear transformation. The dimension of the image of T is equalto
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