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Solving Ax=y by Inspection

Sometimes it is possible to solve a system of linear equations by inspection. This requires some degree of familiarity with the column space of a matrix.

The expression Ax=y denotes saying that y is a linear combination of the columns of A or that equivalently it is in the column space of A. The amounts by which columns of A contribute to the linear combination are determined by the elements of x. Knowing this enables us to solve for x by inspection.

If y is in the column space of A then there is at least one solution; otherwise, there is no solution.

The column space of A is the vector space containing all possible linear combinations of the columns of A.

Here are some examples.

Example 1

        3312 x0 x1 x2 x3   =  14

Here is one possible way of writing y as a linear combination of the columns ci of A:

        14   =  2 c0 +2 c1 +0 c2 +1 c3

        14   =  2 3 +  2 3 +  0 1 +  1 2

        14   =  x0 3 +  x1 3 +  x2 1 +  x3 2

             x0 x1 x2 x3   =  2 2 0 1

The solution x is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

Example 2

        3001 2311 x0 x1 x2 x3   =  5 8

Here is one possible way of writing y as a linear combination of the columns ci of A:

        5 8   =  1 c0 +1 c1 +1 c2 +2 c3

        5 8   =  1 3 2 +  1 0 3 +  1 0 1 +  2 1 1

        5 8   =  x0 3 2 +  x1 0 3 +  x2 0 1 +  x3 1 1

             x0 x1 x2 x3   =  1 1 1 2

The solution x is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

Example 3

        1322 0310 1232 x0 x1 x2 x3   =  8 2 10

Here is one possible way of writing y as a linear combination of the columns ci of A:

        8 2 10   =  2 c0 +0 c1 +2 c2 +1 c3

        8 2 10   =  2 1 0 1 +  0 3 3 2 +  2 2 1 3 +  1 2 0 2

        8 2 10   =  x0 1 0 1 +  x1 3 3 2 +  x2 2 1 3 +  x3 2 0 2

             x0 x1 x2 x3   =  2 0 2 1

The solution x is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

Example 4

        2031 1300 3030 0232 x0 x1 x2 x3   =  8 2 9 5

Here is one possible way of writing y as a linear combination of the columns ci of A:

        8 2 9 5   =  2 c0 +0 c1 +1 c2 +1 c3

        8 2 9 5   =  2 2 1 3 0 +  0 0 3 0 2 +  1 3 0 3 3 +  1 1 0 0 2

        8 2 9 5   =  x0 2 1 3 0 +  x1 0 3 0 2 +  x2 3 0 3 3 +  x3 1 0 0 2

             x0 x1 x2 x3   =  2 0 1 1

The solution x is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

[This content was created with ML, available from the App Store.]