Question

Show that \(\left\\{(a, b),\left(a_{1}, b_{1}\right)\right\\}\) is a basis of \(\mathbb{R}^{2}\) if and only if \(\left\\{a+b x, a_{1}+b_{1} x\right\\}\) is a basis of \(\mathbf{P}_{1}\).

Short answer

\(\{(a, b), (a_1, b_1)\}\) forms a basis of \(\mathbb{R}^2\) if the determinant \(ab_1 - a_1b \neq 0\), ensuring \(\{a+bx, a_1+b_1x\}\) is a basis of \(\mathbf{P}_{1}\).

Step-by-step solution

Understand Basis in \(\mathbb{R}^2\)

A set of vectors \(\{(a, b), (a_1, b_1)\}\) is a basis for \(\mathbb{R}^2\) if the vectors are linearly independent and span \(\mathbb{R}^2\). This is true if the determinant of the matrix formed by these vectors is non-zero.

Calculate Determinant in \(\mathbb{R}^2\)

The vectors form the matrix \(\begin{bmatrix} a & a_1 \ b & b_1 \end{bmatrix}\). The determinant is given by \(ab_1 - a_1b\). This determinant must be non-zero for the vectors to form a basis of \(\mathbb{R}^2\).

Understand Basis in \(\mathbf{P}_{1}\)

A basis for \(\mathbf{P}_{1}\), the space of polynomials of degree at most 1, consists of two polynomials that are linearly independent and span \(\mathbf{P}_{1}\). The polynomials \(a + bx\) and \(a_1 + b_1x\) should satisfy this property.

Check Linear Independence in \(\mathbf{P}_{1}\)

For the polynomials \(a + bx\) and \(a_1 + b_1x\) to be linearly independent, the only solution to \(c_1(a + bx) + c_2(a_1 + b_1x) = 0\) should be \(c_1 = 0\) and \(c_2 = 0\). This equation translates to the same condition as the determinant from Step 2.

Check Spanning in \(\mathbf{P}_{1}\)

Because the polynomials span the space \(\mathbf{P}_{1}\), any polynomial of degree 1, such as \(p(x) = \, q + rx\), must be expressible as a linear combination of \(a + bx\) and \(a_1 + b_1x\). This reflects the same solution as having a non-zero determinant from Step 2.

Conclusion: Equivalence of Bases

Thus, \(\{(a, b), (a_1, b_1)\}\) forms a basis in \(\mathbb{R}^2\) if and only if \(\{a + bx, a_1 + b_1x\}\) forms a basis in \(\mathbf{P}_{1}\) because both conditions depend on the non-zero determinant \(ab_1 - a_1b\).

Key concepts

Basis of a Vector Space

In the world of linear algebra, a "basis" is like a solid foundation of building blocks for a vector space.
Imagine a vector space as a large flat piece of land. The basis would be the set of unique markers you place on this land to define its boundaries. In more technical terms:

• The basis is a set of vectors that are linearly independent.
• They span the entire space.

This means every vector in the space can be expressed as a combination of these basis vectors. If we're talking about \( \mathbb{R}^2 \), a familiar plane,

• We need exactly two vectors in our basis.
• These vectors must cover all directions in the plane.
• They must not lie on the same line.

The condition for a valid basis in \( \mathbb{R}^2 \) is that the determinant of the basis vectors is non-zero. The determinant gives us a way to measure how 'spread out' the vectors are in the space.

Linear Independence

Linear independence is a crucial concept in ensuring that a set of vectors truly works as a basis.
What does it mean? Simply put, if vectors are linearly independent, none of them can be written as a combination of the others. Think of it like having different flavors of ice cream where none can be made from mixing the others.
To check for linear independence, you can use the following method:

• Take your set of vectors.
• Form a matrix with these vectors as columns.
• Find the determinant of this matrix.

If the determinant is not zero, the vectors are linearly independent. This rule is straightforward in \( \mathbb{R}^2 \) because it ensures that neither vector is redundant. In our case, we used the determinant condition \( ab_1 - a_1b eq 0 \) for vectors \( (a, b) \) and \( (a_1, b_1) \) to check independence.

Polynomial Space

The space \( \mathbf{P}_{1} \) is a set of all polynomials of degree at most 1. This means they can look like \( a + bx \), where \( a \) and \( b \) are constants. Even though the space of polynomials seems different from vectors you might visualize in \( \mathbb{R}^2 \), mathematicians treat them similarly.
Similar rules apply:

• The polynomial space needs a basis to be fully defined.
• A basis in this space consists of two linearly independent polynomials.
• These polynomials span the entire \( \mathbf{P}_{1} \) space.

The major link to our problem lies in equivalence. In \( \mathbf{P}_{1} \), forming a basis means we can create any polynomial of form \( q + rx \) using our chosen ones just like forming any vector in \( \mathbb{R}^2 \). For the polynomials \( a + bx \) and \( a_1 + b_1x \) to act as a basis, they must not be multiples of each other, aligning with the linear independence principle.