Find all possible values of a and b in the inner product ⟨𝐯, 𝐰⟩=a v1 w1+b v2 w2 that make the ve

step 1 Hello, so here we consider the vectors B3, 2, and 1B1, which are orthogonal. Then by the definit

Find all possible values of a and b in the inner product ⟨𝐯,...

Find all possible values of $a$ and $b$ in the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=a v_1 w_1+b v_2 w_2$ that make the vectors $(1,2)^T,(-1,1)^T$, an orthogonal basis in $\mathbb{R}^2$.

Key Concepts

Inner Product

An inner product is a function that takes two vectors and returns a scalar, satisfying properties such as linearity, symmetry, and positive definiteness. It provides the framework to introduce geometric concepts like angles and lengths into vector spaces. In problems involving parameterized inner products, understanding these properties is essential to appropriately adjusting the inner product on the space.

Orthogonality

Orthogonality is the condition where the inner product of two vectors equals zero. This concept is critical in vector spaces because it allows for simplification of various computations, such as projections and decompositions. By enforcing that the inner product is zero, one ensures that the vectors are independent in the geometric sense defined by the inner product.

Basis

A basis in a vector space is a set of linearly independent vectors that span the entire space. When a basis is orthogonal with respect to a given inner product, calculations and geometric interpretations become more straightforward. Problems that involve constructing or verifying an orthogonal basis often hinge on choosing or verifying parameters in the inner product accordingly.

Parameterization of Inner Products

In certain cases, inner products can be defined with parameters that weight different components of the vectors, leading to a family of inner products. Choosing these parameters correctly is essential to ensure that the inner product retains its basic properties and that certain vectors meet conditions like orthogonality. This involves setting up equations based on the inner product's definition and solving for the parameters in question.

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