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## Homework Statement

I know this stuff isnt complicated but the definitions of my book are very formal and confusing. I have to find the image and the kernel of these two matrices:

[itex]

A= \begin{bmatrix}1&2&3\end{bmatrix}

B= \begin{bmatrix}2&3\\6&9\end{bmatrix} [/itex]

## The Attempt at a Solution

my book says the image is the span of the column vectors so for A would that be:

[itex] im(A) = x_1[1] + x_2[2] + x_3[3] [/itex] ???

and since 3 = 1 + 2 I think that makes 3 redundant so it would be x1[1] + x2[2]?

and the kernel is when the system = 0 so when

x1 = -2x2 - 3x3

if x2 = t and x3 = r then

[itex] ker(A) = [-2t-3r] ?? [/itex]

for B

[itex]im(B) = x_1 \begin{bmatrix}2\\6\end{bmatrix} + x_2 \begin{bmatrix}3\\9\end{bmatrix} [/itex]

[itex] ker(B) = \begin{bmatrix}2&3&|0\\6&9&|0\end{bmatrix} = \begin{bmatrix}1&3/2&|0\\6&9&|0\end{bmatrix}

=

\begin{bmatrix}1&3/2&|0\\0&0&|0\end{bmatrix} [/itex]

so x1 = -3/2 x2

if x2 = t then

[itex] ker(B) = [-3/2t] [/itex]??

am i doing these right? I would appreciate if someone can help explain these concepts to me without using the formal abstract definitions

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