Question

I need some help with these true false questions for linear algebra:

9. True or False - give a briel explanation of your reasoning WAB 4x matrix with rank 3, then the equation AR -0 has a unique solution b. a linear map :R + R" has nullity, then it is onto C V a span ....is a 3-dimensional vector space, then ... is a basis for d aina Ais diagonarable, then it is also invertible

a. If Ais a 4 x 3 matrix with rank 3, then the equation Ax = 0 has a unique solution. T or F?

b. If a linear map f: R^n goes to R^n has nullity 0, then it is onto. T or F?

c. If V = span{v1, v2, v3,} is a 3-dimensional vector space, then {v1, v2, v3} is a basis for V. T or F?

d. If a matrix A is diagonalizable, then it is also invertible. T or F?

Answer 1

Q Is A is ux3 matain with rank 3, then equation AXTO has unique solution ans It is true, because of rank 3 it means 3 is. number of nonjero rows in reduced row echelon form of a and number of Columns are also 3. it means Axo has only one solution which is trivial (0,0,0) 6). If a linear map f. R "R" has nulitig o, then it is onto tait is true becouse if mility is Jero = f(x)=0 has only trivial solution (2-0) I so by rank mility theorem Y+1=n Cheges is rank off and n is mility of of so Yonk of fan - it is one one and dimension of domain and codomain is ennal so it is onto also an is equal so

if ve spem {v, , V2, Vn} is a. 3-dimensional Vellor spence, then {v, , V2, V3} is a basis I for v. ong Trive, becouse a linear independent spanning set of v. is called basis for here since ris 3-dimensional vector space and spane by 3 vector So all three vector are together make basis for v d) if a matrini A is diagonalizable, then I it is also invertiable cons - It is false, we have a counterexample Consider a jero matrix. The zero matrin is diagonal and this it is diagonalizable. However it is not invertiable becouse determinat is zero.