Question 9 need helpThanks Define the matrix A 1 0 2 c and

Question 9 need help.Thanks

Define the matrix, A = (1 0 2 c), and define the map L: R^2 rightarrow R^2 via matrix multiplication. That is, if v = (x y) R^2, then L is defined as L(v) = Av = (1 0 2 c) (x y). Prove that L is subjective if and only if c notequalto 0.

Solution

Range of L lies in R2

SO L is surjective when range of R has dimension 2

if c=0 then first and second columns are multiples of each other and hence the range of L has dimension 1 if c=0 and L is not surjective

So , c must be non zero

Assume c is non zero

then two columns of the matrix are linearly independent and hence span R2 because R2 has dimension 2