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I am presently going through Strang's linear algebra book in which he explains row picture and column picture for a system of linear equations as such

For example: $$\begin{cases} 2x + 3y = 5 \\ 4x + 7y = 9 \end{cases} $$

are two lines in row picture and in column picture they are 2-d vectors in the 2-D space

similarly, $$ \begin{cases} 2u + v + w = 5 \\ 4u - 6v = -2 \\ -2u + 7v + 2w = 9 \end{cases}$$

in row picture these represent 3 planes ( the second equation still a plane with $w$ taking any value) and from column picture consists 3-d vectors of 3-D space

What can i infer for a system of linear equations with $n$ equations and $m$ unknowns such as this $$ \begin{cases} 2x + 7y = 9 \\ 3x + 8y = 11 \\ 3x - 2y = 4 \\ x + y = 6 \end{cases}$$

what does the row picture shows for this system of linear equations ? Column picture is in 4D space.

I am new to linear algebra, please correct me if my understanding is wrong. Thanks

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2 Answers 2

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You can think that the system represents the intersection of four hyperplanes ( isomorphic to $3-D$ spaces) and the solution $P=(x_0,y_0,t_0,s_0)$ is the common point of these hyperplanes.

The ''columns'' interpretation can be a linear transformation from a $2-D$ space to a $4-D$ space.

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The row picture shows simply four lines; the lines of your example have no common point of intersection, so there is no solution.

The column picture are two vectors in 4-space, plus a vector for the result. A solution can only be found if the result vector happens to lie in the "plane" spanned by the two vectors. (This is easier to visualize if you use three equations instead of four.)

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