Find the basis and dimension of the subspace span
dd by the vectors (2,-3,1), (3,0,1) and (1,1,1)

Question

Find the basis and dimension of the subspace spandd by the vectors (2,-3,1), (3,0,1) and (1,1,1)

Nick Camelot · Accepted Answer

To find the basis and dimension of the subspace spanned by the vectors

(2, - 3, 1)

,

(3, 0, 1)

, and

(1, 1, 1)

, we can follow these steps:
Step 1: Create a matrix using the given vectors as its columns:

[\begin{matrix} 2 & 3 & 1 \\ - 3 & 0 & 1 \\ 1 & 1 & 1 \end{matrix}]

Step 2: Perform row operations to bring the matrix to its reduced row-echelon form or row-reduced echelon form.

[\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix}]

Step 3: Identify the pivot columns in the reduced row-echelon form. These columns correspond to the vectors that form the basis of the subspace.
In this case, the first and second columns have pivot elements.
Step 4: Express the vectors corresponding to the pivot columns as a linear combination.
Let's call the vectors corresponding to the pivot columns

𝐯_{1}

and

𝐯_{2}

.

𝐯_{1} = (2, - 3, 1)

𝐯_{2} = (3, 0, 1)

Step 5: The basis of the subspace is formed by the linearly independent vectors

𝐯_{1}

and

𝐯_{2}

.
The basis of the subspace spanned by the vectors

(2, - 3, 1)

,

(3, 0, 1)

, and

(1, 1, 1)

is

{(2, - 3, 1), (3, 0, 1)}

.
Step 6: The dimension of the subspace is equal to the number of vectors in the basis. In this case, the dimension is 2.

Find the basis and dimension of the subspace

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