# Linear Transformations, Kernel and Image of Matrices

• Aug 26th 2009, 02:13 PM
rak
Linear Transformations, Kernel and Image of Matrices
"Let P2 be the vector space over R of polynomials of degree at most 2, and let F: P2 -> P2 be the function defined by F(p)(x) = p'(x) + 3(p)''(x). Prove that F is a linear transformation. Find the matrix F with respect to the basis {1,x,x^2} of P2. Find bases for Ker(F) and Im(F)."

Right, I can do the first part, showing its a linear transformation, am I right in thinking that to find the matrix I find F(1), F(x) and F(x^2)?

This gives:

( 0 1 6 )
( 0 0 2 )
( 0 0 0 )

Is this correct? Could anybody please tell me how to find the bases for ker(F) and im(F) from here?

Thanks
• Aug 26th 2009, 06:55 PM
NonCommAlg
Quote:

Originally Posted by rak
"Let P2 be the vector space over R of polynomials of degree at most 2, and let F: P2 -> P2 be the function defined by F(p)(x) = p'(x) + 3(p)''(x). Prove that F is a linear transformation. Find the matrix F with respect to the basis {1,x,x^2} of P2. Find bases for Ker(F) and Im(F)."

Right, I can do the first part, showing its a linear transformation, am I right in thinking that to find the matrix I find F(1), F(x) and F(x^2)?

This gives:

( 0 1 6 )
( 0 0 2 )
( 0 0 0 )

Is this correct? Could anybody please tell me how to find the bases for ker(F) and im(F) from here?

Thanks

your matrix is correct. call this matrix A. now identify the polynomial $\displaystyle p(x)=a+bx+cx^2$ with the column $\displaystyle \bold{x}=\begin{pmatrix}a \\ b \\ c \end{pmatrix}.$ then to find the kernel you just need to solve $\displaystyle A \bold{x} = 0.$ that gives you
$\displaystyle b=c=0.$ so a basis for the kernel is $\displaystyle \{ 1 \}.$ now since $\displaystyle A \bold{x} = \begin{pmatrix}b+6c \\ 2c \\ 0 \end{pmatrix}=(b+6c) \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + 2c \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix},$ a basis for the image is $\displaystyle \{1, x \}.$