1)Let T : R³ → R⁴ be given by T(v) =Av where A is any matrix such as 1 0 1

2 0 3

0 1 0

3 4 2

Find the dimension of the null space

Find a basis for the range space of T

2) let V and W be finite dimensional vector spaces and let T : V → W be a linear transformation. Show that dim(ker T) + dim(Im T) = dim(V)

3)let V be the vector space over the field R of real numbers consisting of all functions from R into R.let U be the subspace of even functions and W the subspace of odd functions.show that V=U + W

4)let { v1, v2, ......,vn} be a basis for a vector space V over R. Show that if w is any vector in V, then for some choice of sign ±, {v1 ± w, v2,....., vn} is a basis for V

2 0 3

0 1 0

3 4 2

Find the dimension of the null space

Find a basis for the range space of T

2) let V and W be finite dimensional vector spaces and let T : V → W be a linear transformation. Show that dim(ker T) + dim(Im T) = dim(V)

3)let V be the vector space over the field R of real numbers consisting of all functions from R into R.let U be the subspace of even functions and W the subspace of odd functions.show that V=U + W

4)let { v1, v2, ......,vn} be a basis for a vector space V over R. Show that if w is any vector in V, then for some choice of sign ±, {v1 ± w, v2,....., vn} is a basis for V