How to find the dimension of a subspace?

Understand the Problem

The question is asking how to determine the dimension of a subspace in a vector space. To solve this, we typically look for a basis for the subspace and count the number of vectors in that basis, as the dimension is defined as the number of vectors in a basis.

Answer

The dimension of the subspace is determined by the number of linearly independent vectors in the set.

Steps to Solve

Identify the subspace and its vectors

Begin by identifying the vectors that define the subspace. They should be represented in a vector format, such as $v_1, v_2, ..., v_n$.

Form a matrix from the vectors

Construct a matrix by placing the vectors as rows or columns. For example, if the vectors are placed as rows, the matrix would be:

$$ A = \begin{bmatrix} v_1 \ v_2 \ \vdots \ v_n \end{bmatrix} $$

Row reduce the matrix

Perform row reduction (Gaussian elimination) on the matrix to bring it to row echelon form. This will help to visualize which rows are linearly independent.

Count the number of leading 1's

Identify the number of leading 1's (or pivot positions) in the row-reduced matrix. Each leading 1 corresponds to a linearly independent vector in the subspace.

Determine the dimension

The dimension of the subspace is equal to the number of leading 1's found in step 4. Therefore, if there are $k$ leading 1's, the dimension of the subspace is $k$.

The dimension of the subspace is equal to the number of linearly independent vectors found in the given set.

More Information

The dimension is a fundamental concept in linear algebra as it gives us insight into the size of a vector space. A larger dimension indicates more degrees of freedom within the space.

Tips

Confusing the total number of vectors with the number of linearly independent vectors. Always check for linear independence.
Failing to perform the full row reduction and overlooking additional pivot positions.

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