Examples
[3-1021-1]⎡⎢⎣3−1021−1⎤⎥⎦
Step 1
Step 1.1
Multiply each element of R1R1 by 1313 to make the entry at 1,11,1 a 11.
Step 1.1.1
Multiply each element of R1R1 by 1313 to make the entry at 1,11,1 a 11.
[33-13021-1]⎡⎢
⎢⎣33−13021−1⎤⎥
⎥⎦
Step 1.1.2
Simplify R1R1.
[1-13021-1]⎡⎢
⎢⎣1−13021−1⎤⎥
⎥⎦
[1-13021-1]⎡⎢
⎢⎣1−13021−1⎤⎥
⎥⎦
Step 1.2
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Step 1.2.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[1-13021-1-1+13]⎡⎢
⎢⎣1−13021−1−1+13⎤⎥
⎥⎦
Step 1.2.2
Simplify R3R3.
[1-13020-23]⎡⎢
⎢⎣1−13020−23⎤⎥
⎥⎦
[1-13020-23]⎡⎢
⎢⎣1−13020−23⎤⎥
⎥⎦
Step 1.3
Multiply each element of R2R2 by 1212 to make the entry at 2,22,2 a 11.
Step 1.3.1
Multiply each element of R2R2 by 1212 to make the entry at 2,22,2 a 11.
[1-1302220-23]⎡⎢
⎢
⎢⎣1−1302220−23⎤⎥
⎥
⎥⎦
Step 1.3.2
Simplify R2R2.
[1-13010-23]⎡⎢
⎢⎣1−13010−23⎤⎥
⎥⎦
[1-13010-23]⎡⎢
⎢⎣1−13010−23⎤⎥
⎥⎦
Step 1.4
Perform the row operation R3=R3+23R2R3=R3+23R2 to make the entry at 3,23,2 a 00.
Step 1.4.1
Perform the row operation R3=R3+23R2R3=R3+23R2 to make the entry at 3,23,2 a 00.
[1-13010+23⋅0-23+23⋅1]⎡⎢
⎢⎣1−13010+23⋅0−23+23⋅1⎤⎥
⎥⎦
Step 1.4.2
Simplify R3R3.
[1-130100]⎡⎢
⎢⎣1−130100⎤⎥
⎥⎦
[1-130100]⎡⎢
⎢⎣1−130100⎤⎥
⎥⎦
Step 1.5
Perform the row operation R1=R1+13R2R1=R1+13R2 to make the entry at 1,21,2 a 00.
Step 1.5.1
Perform the row operation R1=R1+13R2R1=R1+13R2 to make the entry at 1,21,2 a 00.
[1+13⋅0-13+13⋅10100]⎡⎢
⎢⎣1+13⋅0−13+13⋅10100⎤⎥
⎥⎦
Step 1.5.2
Simplify R1R1.
[100100]⎡⎢⎣100100⎤⎥⎦
[100100]⎡⎢⎣100100⎤⎥⎦
[100100]⎡⎢⎣100100⎤⎥⎦
Step 2
The pivot positions are the locations with the leading 11 in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: a11a11 and a22a22
Pivot Columns: 11 and 22
Step 3
The basis for the column space of a matrix is formed by considering corresponding pivot columns in the original matrix. The dimension of Col(A)Col(A) is the number of vectors in a basis for Col(A)Col(A).
Basis of Col(A)Col(A): {[301],[-12-1]}⎧⎪⎨⎪⎩⎡⎢⎣301⎤⎥⎦,⎡⎢⎣−12−1⎤⎥⎦⎫⎪⎬⎪⎭
Dimension of Col(A)Col(A): 22