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**For each matrix A, find vectors that span the image of A. Give as few vectors as possible.**

[tex]\mathbf{A_1} =

\left[ \begin{array}{cc}

1 & 1 \\

1 & 2 \\

1 & 3 \\

1 & 4

\end{array} \right][/tex]

[tex]\mathbf{A_2} =

\left[ \begin{array}{cccc}

1 & 1 & 1 & 1 \\

1 & 2 & 3 & 4 \\

\end{array} \right][/tex]

__My work__:

[tex]T(\overrightarrow{x})\,=\,A_1\,\overrightarrow{x}[/tex]

[tex]A_1\,\left[ \begin{array}{c}

x_1 \\

x_2 \\

x_3 \\

x_4

\end{array} \right]\,=\,\left[ \begin{array}{cc}

1 & 1 \\

1 & 2 \\

1 & 3 \\

1 & 4

\end{array} \right]\,\left[ \begin{array}{c}

x_1 \\

x_2

\end{array} \right][/tex]

[tex] x_1\,\left[ \begin{array}{c}

1 \\

1 \\

1 \\

1

\end{array} \right]\,+\,x_2\,\left[ \begin{array}{c}

1 \\

2 \\

3 \\

4

\end{array} \right][/tex]

The image of [tex]A_1[/tex] is the space spanned by

[tex]v_1\,=\,\left[ \begin{array}{c}

1 \\

1 \\

1 \\

1

\end{array} \right]\,\,and\,\,v_2\,=\,\left[ \begin{array}{c}

1 \\

2 \\

3 \\

4

\end{array} \right][/tex]

For the second part:

[tex]A_2\,\overrightarrow{x}[/tex]

[tex]\left[ \begin{array}{cccc}

1 & 1 & 1 & 1 \\

1 & 2 & 3 & 4 \\

\end{array} \right]\,\left[ \begin{array}{c}

x_1 \\

x_2 \\

x_3 \\

x_4

\end{array} \right][/tex]

[tex]x_1\,\left[\begin{array}{c}

1 \\

1 \\

\end{array}\right]\,+\,x_2\,\left[\begin{array}{c}

1 \\

2 \\

\end{array}\right]\,+\,x_3\,\left[\begin{array}{c}

1 \\

3 \\

\end{array}\right]\,+\,x_4\,\left[\begin{array}{c}

1 \\

4 \\

\end{array}\right][/tex]

But now I don't know where to proceed to solve the second part. The B.O.B. says that the answer is

[tex]\left[\begin{array}{c}

1 \\

1 \\

\end{array}\right],\,\left[\begin{array}{c}

1 \\

2 \\

\end{array}\right][/tex]

How do I show such?

Last edited: