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A quick example calculating the column space and the nullspace of a matrix. Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences Fig from University of Iowa knotplot.com

Column space of A = col A =

col A = span , , ,

Column space of A = col A =

col A = span , , ,

= c

1

+ c

2

+ c

3

+ c

4

c

i

in R

Put A into echelon form: R 2

• R 1  R 2 R 3

• 2R 1  R 3

Put A into echelon form: And determine the pivot columns R 2

• R 1  R 2 R 3

• 2R 1  R 3

Put A into echelon form: And determine the pivot columns R 2

• R 1  R 2 R 3

• 2R 1  R 3

Put A into echelon form: A basis for col A consists of the 3 pivot columns from the original matrix A. Thus basis for col A = R 2

• R 1  R 2 R 3

• 2R 1  R 3

A basis for col A consists of the 3 pivot columns from the original matrix A. Thus basis for col A = Note the basis for col A consists of exactly 3 vectors. Thus col A is 3-dimensional.

col A contains all linear combinations of the 3 basis vectors:

col A = c

1

+ c

2

+ c

3

c

i

in R

col A contains all linear combinations of the 3 basis vectors:

col A = c

1

+ c

2

+ c

3

c

i

in R

= span , ,

Can you identify col A?

Determine the nullspace of A

Put A into echelon form and then into reduced echelon form: R 2

• R 1  R 2 R 3

• 2R 1  R 3 R 1
• 5R 2  R 1 R 2 /2  R 2 R 1
• 8R 3  R 1 R 1

• 2R 3  R 1 R 3 /3  R 3

Solve: A x = 0 where A =

Put A into echelon form and then into reduced echelon form: R 2

• R 1  R 2 R 3

• 2R 1  R 3 R 1
• 5R 2  R 1 R 2 /2  R 2 R 1
• 8R 3  R 1 R 1

• 2R 3  R 1 R 3 /3  R 3

Solve: A x = 0 where A

x 1 x 2 x 3 x 4 0 0 0 x 1

• 2 x 4

x 2

• 2 x 4

x 3

• x 4

x 4 x 4

x 4