How can we verify if a set of vectors spans a subspace?

To verify if a set of vectors spans a subspace, you need to check if every vector in the subspace can be written as a linear combination of the vectors in the given set. Here is a step-by-step process to determine if a set of vectors spans a subspace:

### 1. **Write the vectors as a matrix**:
   Arrange the given vectors as columns in a matrix. For example, if you are given \( \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \), you can form a matrix \( A = [ \mathbf{v}_1 \; \mathbf{v}_2 \; \dots \; \mathbf{v}_n ] \).

### 2. **Check linear independence**:
   A set of vectors spans a subspace if they are linearly independent and their span equals the dimension of the subspace. To check for linear independence, you can either:
   - **Row-reduce the matrix** to its row echelon form (or reduced row echelon form). If the number of non-zero rows in the row-reduced matrix equals the number of vectors in the set, the vectors are linearly independent.
   - Alternatively, calculate the **determinant** (only for square matrices) to verify if it's non-zero, indicating independence.

### 3. **Determine the rank**:
   After row-reducing the matrix, the rank (number of non-zero rows) tells you how many linearly independent vectors you have. If the rank equals the number of vectors, they are linearly independent. If the rank equals the dimension of the subspace, the set of vectors spans the subspace.

### 4. **Ensure the dimension matches**:
   For the vectors to span the subspace, the number of independent vectors must match the dimension of the subspace you're trying to span. For example, in \( \mathbb{R}^3 \), you need 3 linearly independent vectors to span the entire space.

### 5. **Verify if all vectors in the subspace can be expressed as linear combinations**:
   If the set of vectors is linearly independent and the rank matches the subspace's dimension, you can conclude that the vectors span the subspace.

### Example:

If you're given vectors \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \), these vectors form the identity matrix, which spans \( \mathbb{R}^3 \).

### Conclusion:
The set of vectors spans a subspace if:
1. They are linearly independent.
2. The number of independent vectors equals the dimension of the subspace.

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### Related Questions:
1. How can you row-reduce a matrix to check for independence?
2. What is the significance of the rank of a matrix in linear algebra?
3. How do you determine if a set of vectors forms a basis for a vector space?
4. Can dependent vectors span a subspace?
5. What is the difference between spanning a subspace and spanning the entire space?

### Tip:
A set of vectors in \( \mathbb{R}^n \) can span the whole space \( \mathbb{R}^n \) only if there are exactly \( n \) linearly independent vectors in the set.

Discover how to verify if a set of vectors spans a subspace using linear algebra principles. Learn about matrix rank, linear independence, and row-reduction techniques to determine if vectors form a basis for the subspace. Suitable for undergraduate students.

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