Question

Determine, with justification, whether each of the following statements is true or false. (a) IfV is a vector space and S, and S2 are two bases of V, then Si U S2 is a basis of V. (b) Let A and B ne matrices of the same size. If A and B have the same row space, then they have the same column space. (c) Let M be an n x n square matrix. If M has less than n distinct eigenvalues, then M is not diagonalisable.

Answer 1

(a)False

let us take example , V=$R^{2}$, their basis S1={(1,0), (0,1)} and S2={(1,1), (1,-1)}.

Now, ={(1,0),(0,1), (1,1), (1,-1)} is not a basis of V.

SUS

(b) True

Here A and B are matrices of same size and have the same row space that implies A and B have same rank.

Dimension of coloumn space of the matrix is also equal to the rank of that matrix.

So, if A and B have the same row space then their coloumn space is also same.

(c) False

let us take a example , M= $\begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix}$ be a square order matrix.

Their eigenvalue is 1,1.

No. of distinct eigenvalue is 1 which is less than 2.

characterstics polynomial is $c(x)=\left ( x-1 \right )^{2}$ and minimal polynomial is$m(x)=(x-1)$.

Here, minimal polynomial is linear factor and we know that if minimal polynomial has linear factor then the matrix is diagonalisable.