Linear Algebra Answers

Let T:R^3--->R^4 be defined by T(x1,x2,x3)=(x1+x2, x2+x3, x1-x3, 2x1+x2-x3). Check T is operator. Find kernel and range of T. Find dimensions of kernel.

Let U,V be subspaces of R^n.

Show that U∩V={0} if and only if S∪T is a linearly independent set of vectors for every linearly independent set S={u1,u2,...uk} ⊆ U and every linearly independent set T = {v1,v2,...vl} ⊆ V

Let V be the solution space of the following homogeneous linear system: x1 − x2 − 2x3 + 2x4 − 3x5 = 0 x1 − x2 − x3 + x4 − 2x5 = 0. (a) (2 points) Find a basis S of V and write down the dimension of V.

(b) (3 points) Finda subspace W of R5 suchthat W contains V anddim(W) = 4. Justify your answer.

Let V be a subspace of R^4 and S = {u1,u2,u3} be a basis for V. Suppose v1,v2,v3 are vectors in V such that(v1)S = (1,−2,0), (v2)S =(2,−7,4),and (v3)S =(−3,8,−1).

Suppose v1 = (5,−5,0,0), v2 = (10,5,−10,−10), and v3 =(−5,0,−5,5).

Find u1, u2, and u3.

Let V be a subspace of R^4 and S = {u1,u2,u3} be a basis for V. Suppose v1,v2,v3 are vectors in V such that(v1)S = (1,−2,0), (v2)S = (2,−7,4), and (v3)S = (−3,8,−1).

Show that {v1, v2, v3} is also a basis for V.

Determine whether the set S of vectors is linear independent or not: S = {u1, u2, u3, u4} ⊆ R^4, where u1, u2, u3, u4 are all different and it is known that (1, 0, 0, 0) is not a member of

[img]https://upload.cc/i1/2020/03/13/UfFVyi.jpg[/img]




R is 4

Let V be a subspace of R^4 and S = {u1,u2,u3} be a basis for V. Suppose v1, v2, v3 are vectors in V such that (v1)S = (1,−2,0), (v2)S =(2,−7,4), and (v3)S =(−3,8,−1).

Show that {v1, v2, v3} is also a basis for V.

[img]https://upload.cc/i1/2020/03/12/Bl1aVO.jpg[/img]




R is 4

Let U,V be subspaces of Rn. Show that U ∩ V = {0} if and only if S ∪ T is a linearly independent set of vectors for every linearly independent set S = {u1,u2,...,uk} ⊆ U and every linearly independent set T = {v1,v2,...,vl} ⊆ V.