LINEAR ALGEBRA: Find vectors that span the image of A

[VinnyCee]

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

$$\mathbf{A_1} = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \end{array} \right]$$


$$\mathbf{A_2} = \left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ \end{array} \right]$$


My work:

$$T(\overrightarrow{x})\,=\,A_1\,\overrightarrow{x}$$

$$A_1\,\left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right]\,=\,\left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \end{array} \right]\,\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]$$

$$x_1\,\left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array} \right]\,+\,x_2\,\left[ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right]$$

The image of $$A_1$$ is the space spanned by

$$v_1\,=\,\left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array} \right]\,\,and\,\,v_2\,=\,\left[ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right]$$

For the second part:

$$A_2\,\overrightarrow{x}$$

$$\left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ \end{array} \right]\,\left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right]$$

$$x_1\,\left[\begin{array}{c} 1 \\ 1 \\ \end{array}\right]\,+\,x_2\,\left[\begin{array}{c} 1 \\ 2 \\ \end{array}\right]\,+\,x_3\,\left[\begin{array}{c} 1 \\ 3 \\ \end{array}\right]\,+\,x_4\,\left[\begin{array}{c} 1 \\ 4 \\ \end{array}\right]$$

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

$$\left[\begin{array}{c} 1 \\ 1 \\ \end{array}\right],\,\left[\begin{array}{c} 1 \\ 2 \\ \end{array}\right]$$

How do I show such?

 

[HallsofIvy]

You are trying to find a basis for the "row space". Treat each row as a vector and reduce. Obviously you have 4 rows, so 4 vectors, but the image is subset of R² and so can have dimension no larger than 2.

Actually, if the dimension is 2, then (1,0), (0, 1) will work. If 1, anyone of the rows will work.

 

[VinnyCee]

$$A_1$$ has four rows, but only two vectors. I am assuming that the number of columns should equal the number of vectors?

Where are you getting the (1, 0) and (0, 1 ) from?

By "dimension", do you mean the number of columns in a matrix, or the number of rows?

 

[bowzerman]

Leading ones.

Solve for leading ones. you will get something like: [(1,0,-1,-2);(0,1,2,3)]. You can see that the columns with leading ones refer you back to the original matrix, which gives you the answer that the BOB had. Hope that helps.