Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) =...
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
Q4. Let T :V + W be a one-to-one linear transformation (i.e. an equation T(u) = T(v) always implies u = v). (a) Show that the kernel of T contains only the zero vector. (b) Show that if the set {T(v1),...,T(Un)} is linearly dependent, then the set {01, ..., Un} is linearly dependent as well. Hint: use part (a).
/3pts Let φ : v → w be an linear transformation. Show that φ is one to one if and only if the kernel of φ contains only the zero vector in V.
Let T: V + W be a linear transformation. Assume that T is one-to-one. Prove that if {V1, V2, V3} C V is a linearly independent subset of V, then {T(01), T(v2), T(13)} C W is a linearly independent subset of W.
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.
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...