1 TrueFalse provide lustification for possible partial credi

1 True/False, provide lustification for possible partial credit a) If the vectors v v vi vi span then vi v v is a basis of C b) If the rank of a 7 x 10 matro A is four, then the pace of A must be six dimensional irv is the set of all 2 x 2 matricesAsuch that the vector is in C (A), then V is a subspace of d) There is an invertible 3 x 3 matrix A such that A is the zero matrik e) The composition of an injection and a surjection is always a biection

Solution

a) True-because given it\'s span of C3, and there are 4 vectors taken hence they are basis i.e linearly dependent

b) False-the null space of matrix should always be equal to the rank Hence it is false

c) False-not a subspace of C2,since it is not closed under scalar multiplication

d)false-given A^2=0,Apply det on both the sides then,|A^2|=0 according to the properties this can be written as|A|^2=0,this implies |A|=0,which is non invertible matrix.hence given statement is wrong.