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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,...
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.
Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is 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 {V1, ..., Un} is linearly dependent as well. Hint: use part (a).
6. Given the points A = (0,0), B = (5,1), C = (2,6) on the plane. Use a determinant to find the area of the triangle ABC. 7. Let T:V - W be a linear transformation, and let V1, V2, ..., Un be vectors in V. Suppose that T(vi), T(02),...,T(un) are linearly independent. Show that V1, V2, ..., Vn are linearly independent. 3. Given that 8 - ...) is a basis for a vector space V. Determine if 3 -...
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
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).
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