Problem 3 - Algebra and Geometry

Question

Suppose that \(\mathbf{u}=s_{1} \mathbf{i}+s_{2} \mathbf{j}\) and \(\mathbf{v}=t_{1} \mathbf{i}+t_{2} \mathbf{j}\), where \(s_{1}, s_{2}, t_{1}\) and \(t_{2}\) are real numbers. Find a necessary and sufficient condition on these real numbers such that every vector in the plane of \(\mathbf{i}\) and \(\mathbf{j}\) can be expressed as a linear combination of the vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Step-by-Step Solution

Answer

The condition is that \(s_1t_2 - s_2t_1 \neq 0\).

Step 1: Understand the Problem

We need to find a condition under which any vector in the plane (expressed using \(\mathbf{i}\) and \(\mathbf{j}\)) can be written as a linear combination of vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Step 2: Concept of Linear Independence

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be used to express any vector in the plane if and only if \(\mathbf{u}\) and \(\mathbf{v}\) are linearly independent.

Step 3: Determine Linear Independence

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are linearly independent if their determinant is non-zero. This means we need to ensure that \(\begin{vmatrix} s_1 & t_1 \ s_2 & t_2 \end{vmatrix} eq 0\).

Step 4: Calculate the Determinant

Compute the determinant of the matrix formed by \(\mathbf{u}\) and \(\mathbf{v}\): \[\text{det} = s_1t_2 - s_2t_1.\]

Step 5: Establish the Condition

For \(\mathbf{u}\) and \(\mathbf{v}\) to span the plane, the determinant must be non-zero: \[s_1t_2 - s_2t_1 eq 0.\] This is the necessary and sufficient condition.

Key Concepts

Linear Combination

A linear combination involves expressing a vector as the sum of scalar multiples of other vectors. For example, in a two-dimensional space defined by the basis vectors \(\mathbf{i}\) and \(\mathbf{j}\), any vector \(\mathbf{a}\) can be expressed as \(a_1 \mathbf{i} + a_2 \mathbf{j}\). Similarly, if you have vectors \(\mathbf{u}\) and \(\mathbf{v}\), you can express any vector \(\mathbf{w}\) in the form of \(c_1 \mathbf{u} + c_2 \mathbf{v}\), where \(c_1\) and \(c_2\) are scalars.

This concept is fundamental in understanding how vectors span a space. If every vector in that space can be expressed this way, then you know that your set of vectors is indeed spanning the space. For \(\mathbf{u}\) and \(\mathbf{v}\) to form a basis of the plane, every possible vector in that plane must be representable as their linear combination.

Determinant

The determinant is a special number that can be computed from a square matrix. In this context, the matrix formed by vectors \(\mathbf{u}\) and \(\mathbf{v}\) will tell us a lot about the space they form.

For a 2x2 matrix: \[\begin{vmatrix} s_1 & t_1 \ s_2 & t_2 \end{vmatrix}\], the determinant is calculated using the formula \(s_1t_2 - s_2t_1\). If this determinant is non-zero, it indicates that \(\mathbf{u}\) and \(\mathbf{v}\) are linearly independent. This verifies that \(\mathbf{u}\) and \(\mathbf{v}\) can span the entire plane, since it means they do not lie on the same line.

The determinant being zero would mean that the vectors are linearly dependent and cannot span the plane, as they will only form a one-dimensional line.

Vector Space

A vector space is a set of vectors where you can add any two vectors and multiply vectors by scalars, and the result still belongs to the same set. It is a collection of vectors and includes the concept of direction and magnitude.

In the case of \(\mathbf{u}\) and \(\mathbf{v}\), they form the vector space spanned by the plane containing \(\mathbf{i}\) and \(\mathbf{j}\). In this space, any vector can conceptually be represented as a combination of \(\mathbf{u}\) and \(\mathbf{v}\), provided they are linearly independent. Thus, ensuring that our set of vectors spans the space.

• The plane spanned by \(\mathbf{i}\) and \(\mathbf{j}\) is a two-dimensional vector space.
• If \(\mathbf{u}\) and \(\mathbf{v}\) are independent and can generate any vector in this space, they form a basis for the plane.

The essential property of a basis is that it is the minimal set of vectors needed to span the entire space, and it embodies all the combinations possible in the vector space.

Span

The span of vectors \(\mathbf{u}\) and \(\mathbf{v}\) refers to all possible vectors you can create with linear combinations of \(\mathbf{u}\) and \(\mathbf{v}\). Essentially, the span indicates the range of space that can be 'covered' or 'reached' by these vectors.

To visualize, imagine extending lines in all directions using the given vectors. If the span forms a whole plane, \(\mathbf{u}\) and \(\mathbf{v}\) are capable of representing each point on that plane.

Factors affecting span include:

• Linear independence: a non-zero determinant indicates the vectors span the entire plane.
• Dimension: when discussing \(\mathbf{i}\) and \(\mathbf{j}\), spanning means controlling every direction in the plane.

Hence, when \(s_1t_2 - s_2t_1 eq 0\), it confirms that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) span the entire two-dimensional space, filling the entire breadth of the plane defined by \(\mathbf{i}\) and \(\mathbf{j}\).