91Ó°ÊÓ

When we talk about the "span" of a set of vectors, we're referring to all possible vectors you can create using linear combinations of those vectors. Think of it like a game where you mix together certain amounts of vector ingredients to make something new. If a vector is in the span of other vectors, you can build it from them.

In the original exercise, we needed to find if \( \mathbf{v}_3 \) belongs to the span of \( \{\mathbf{v}_1, \mathbf{v}_2\} \). The solution involves finding values that can combine \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) to result in \( \mathbf{v}_3 \).

• To do this, see if there are numbers (let’s call them scalars) \( c_1 \) and \( c_2 \) such that \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 = \mathbf{v}_3 \).
• If you find such numbers, it means \( \mathbf{v}_3 \) is in the span of the other two vectors.

In our example, after solving an equation system, we found that this occurs when \( h = -6 \). This means with this value, \( \mathbf{v}_3 \) can be constructed from \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. Imagine in a team where each member has their own unique role. If you can replace one member with combinations of the others, they aren't really needed.

If vectors are linearly independent, removing one still leaves you with a completely unique set that can't recreate the missing vector. In linear algebra, this concept ensures that each vector adds new and necessary directions or dimensions.

When the determinant of a matrix of these vectors is nonzero, it signals independence. In the exercise analysis, our focus was to spot cases of dependence instead, but linear independence can be easily inferred when \( h eq 0 \), since that's when our set has unique contributions from each vector.

In contrast to linear independence, linear dependence in a set of vectors means one vector can be expressed as a combination of others. It's like having redundant parts; remove one, and the others can still "cover" for it.

In the exercise, we established that the set \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} \) is linearly dependent when \( h = 0 \). Here's what happens:

• We checked the determinant of a matrix made from these vectors. A zero determinant implies linear dependence.
• Mathematically, it means the vectors lay on top of each other in some dimensions, making one vector unnecessary.

For \( h = 0 \), the vectors overlap in a way that removes one distinct direction, showing that at least one vector is an unnecessary replica of the structure provided by others.

A linear combination involves multiplying each vector by a scalar (which is just a fancy word for a constant number) and then adding them together. It's like creating a mix, where each ingredient (or vector) is used in certain amounts to create a new dish (or vector).

In the original exercise, we tried to see if \( \mathbf{v}_3 \) could be written as a linear combination of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). This requires finding the scalars \( c_1 \) and \( c_2 \) that make the equation \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 = \mathbf{v}_3 \) true.

Here's how it typically works:

• Identify the target vector you want to "make" using other vectors.
• Determine the right "recipe" of scalars that will combine to form the target vector.

Successful linear combinations validate that the resulting vector can be constructed from the original set. In our example, for \( h = -6 \), the right recipe was solved to be \( c_1 = 4 \) and \( c_2 = 1 \).