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Understanding a linear combination is key to grasping how vectors can generate a space. A linear combination of vectors involves creating a new vector by multiplying each original vector by a corresponding scalar (a number) and then adding the results. For example, if you have vectors:

• a = (1,1,0),
• b = (1,0,1),
• c = (0,1,1),

you can multiply them by scalars a, b, and c, respectively. This gives: \(a(1,1,0) + b(1,0,1) + c(0,1,1)\).

When you add these together, the result is a new vector in the form \((x,y,z)\). If any vector in your space, such as \((x,y,z)\), can be written in this form, then the original vectors are said to generate the space \(F^3\). This concept is fundamental in linear algebra as it lays the foundation for understanding vector spaces.

The matrix determinant is a vital tool when determining whether a group of vectors can span a space. To find out if the given vectors generate \(F^3\), we use these vectors to create a matrix: \[\begin{bmatrix}1 & 1 & 0\1 & 0 & 1\0 & 1 & 1\end{bmatrix}\] The determinant of this matrix helps us understand if a unique solution exists for a linear equation formed by these vectors. In our example, calculating the determinant involves a straightforward mathematical process: \[= 1 \cdot (0 \cdot 1 - 1 \cdot 1) - 1 \cdot (1 \cdot 1 - 0 \cdot 1) + 0 \cdot (1 \cdot 1 - 0 \cdot 1)\] This simplifies to \[= -1 - 1 = -2\]Since the determinant is non-zero (\(-2\)), it indicates that the matrix is invertible, meaning our vectors can indeed generate the space \(F^3\). This is because a non-zero determinant signifies that the linear equation formed by the vectors has a unique solution for every possible vector in the space.

Linear equations are central to solving problems in vector spaces. They describe relationships where variables are only multiplied by constants and added together. In this context, we used the equation \[\begin{bmatrix}1 & 1 & 0\1 & 0 & 1\0 & 1 & 1\end{bmatrix} \begin{bmatrix}a \b \c\end{bmatrix}= \begin{bmatrix}x \y \z\end{bmatrix}\]to demonstrate how a combination of three vectors \((1,1,0), (1,0,1), (0,1,1)\) can generate any vector \((x,y,z)\) in \(F^3\).

Solving this linear equation involves finding values for \(a\), \(b\), and \(c\). If we can find these values for every \((x,y,z)\), then the vectors can generate the space. Thanks to the determinant we calculated, we know there is a unique solution for every possible \((x,y,z)\), thus confirming the ability of these vectors to span the space. Understanding how to set up and solve these equations helps to exhibit the power and beauty of linear algebra.