GRE Mathematics: Linear algebra review questions Let T be a linear transformatio

Question

GRE Mathematics: Linear algebra review questions Let T be a linear transformation from a vector space V with dimension 11 onto a vector space W with dimension 7. What is the dimension of the nullspace of T? a) 0 b) 2 c) 3 d) 4 e) 5 Let p (lambda) be the characteristic polynomial for an nxn matrix A. Suppose that 0 is a root of p (lambda). Which statements are always FALSE? I) A has n linearly independent eigenvectors. II) Det (p (A)) = 0 III) A is invertible. a) I only b) II only c) III only d) I and II only e) I and III only

Explanation / Answer

14. option d is the correct answer

vector space of dimmension 11 onto a vector of dimmension 7

dim (null space of T) = number of free variables in T

= 11-7

= 4

15. option b is the correct answer

because det does not equal to zero if one root is zero

remaining statements are true

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