What Are the Bases for Column and Null Spaces of These Matrices?

[aidandeno]

Please help me with these three questions. I'm really struggling to understand these concepts and I think that with an understanding of these three, I will be able to tackle the rest before my test on Wednesday.

Thank you.
http://www.texpaste.com/n/g4rwmzzw
1) $$ A = \left[\begin{matrix}
-6 & -4 & -1 \\
-4 & 6 & -5 \\
-6 & -4 & -1 \\
10 & -2 & 6
\end{matrix}\right] $$
Find a basis for the image of $A$ (or, equivalently, for the linear transformation $T(x)=Ax$) in the form

$$\{ \left[\begin{matrix}
a \\ b \\ c \\ d
\end{matrix}\right] , \left[\begin{matrix}
e \\ f \\ g \\ h
\end{matrix}\right] \}$$

2)$$ A = \left[\begin{matrix}
16 & 0 \\
-8 & 0
\end{matrix}\right] $$
Find bases for the kernel and image of $T(\vec{x}) = A \vec{x}. $

A basis for the kernel of $A$ is $$\{ \left[\begin{matrix}
a\\
b
\end{matrix}\right] \}$$
A basis for the image of $A$ is $$\{ \left[\begin{matrix}
c\\
d
\end{matrix}\right] \}$$

3)
Let $V=\mathbb{R}^{2\times 2}$ be the vector space of $2\times 2$ matrices and let $L :V\to V$ be defined by $L(X) = \left[\begin{array}{cc} 10 &2\cr 20 &4 \end{array}\right] X$

A basis for $\ker (L )$ is:$$\{ \left[\begin{matrix}
a & b\\
c & d
\end{matrix}\right] , \left[\begin{matrix}
e & f\\
g & h
\end{matrix}\right]\}$$

A basis for $\text{ran}(L)$ is:$$\{ \left[\begin{matrix}
i & j\\
k & l
\end{matrix}\right] , \left[\begin{matrix}
m & n\\
o & p
\end{matrix}\right]\}$$

 

[Jameson]

Hi aidandeno,

Welcome to MHB! :)

Please ask one question per thread and show us what you've done so we know where to help you. Let's do #1.

A basis for the image is a basis for the column space of $A$. What is the rank of $A$? How can we find out the number of linearly independent columns?