Abstract Vector Space Question

• epkid08
In summary: The inner product is used in this problem to find the basis for the orthogonal complement of Span(g_1, g_2). It is used to find the values of a, b, and c that make the dot product equal to 0. This will give you the basis for the orthogonal complement.
epkid08

Homework Statement

Let $g_1(t) = t - 1$ and $g_2(t)= t^2+t$. Using the inner product on $P_2$ defined in example 10(b) with $t_1=-1,t_2=0,t_3=1$, find a basis for the orthogonal complement of $Span(g_1, g_2)$.

Homework Equations

From example 10(b)
$\langle p, q \rangle = \sum_{i=1}^{k+1} p(t_i)q(t_i)$

$(p,q)\in P_2$

The Attempt at a Solution

Well the orthogonal complement of $span(g_1,g_2)$ will be $x$ such that $x\cdot (c_1g_1 + c_2g_2) = 0$, but how can I find the basis for that set? And why and where do we need to use inner product? I am confused

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epkid08 said:

Homework Statement

Let $g_1(t) = t - 1$ and $g_2(t)= t^2+t$. Using the inner product on $P_2$ defined in example 10(b) with $t_1=-1,t_2=0,t_3=1$, find a basis for the orthogonal complement of $Span(g_1, g_2)$.

Homework Equations

From example 10(b)
$\langle p, q \rangle = \sum_{i=1}^{k+1} p(t_i)q(t_i)$

$(p,q)\in P_2$

The Attempt at a Solution

Well the orthogonal complement of $span(g_1,g_2)$ will be $x$ such that $x\cdot (c_1g_1 + c_2g_2) = 0$, but how can I find the basis for that set?
Start with this equation. Here, x $\in$ P2, so you can write it as a + bt + ct2.

Where you show the dot product in the equation above, you should be using the inner product from example 10b. Also, put in the formulas for the functions g1 and g2.

epkid08 said:
And why and where do we need to use inner product? I am confused

1. What is an abstract vector space?

An abstract vector space is a mathematical concept that represents a collection of objects, called vectors, that can be added together and multiplied by numbers. It is a generalization of the familiar concept of geometric vectors in physics and is used in various fields such as linear algebra, functional analysis, and physics.

2. How is an abstract vector space different from a geometric vector space?

An abstract vector space is more general than a geometric vector space. In a geometric vector space, vectors have a specific direction and magnitude, while in an abstract vector space, the vectors can have any number of dimensions and do not necessarily have a geometric interpretation. Additionally, the operations of addition and scalar multiplication in an abstract vector space follow certain axioms, while in a geometric vector space, these operations have specific geometric interpretations.

3. What are the axioms that define an abstract vector space?

The axioms that define an abstract vector space are closure under addition, closure under scalar multiplication, commutativity of addition, associativity of addition, existence of a zero vector, existence of additive inverses, distributivity of scalar multiplication over addition, and associativity of scalar multiplication. These axioms ensure that the operations of addition and scalar multiplication behave in a consistent and predictable manner in an abstract vector space.

4. What are some examples of abstract vector spaces?

Some examples of abstract vector spaces include the space of all real-valued functions, the space of all polynomials, and the space of all sequences. These spaces are abstract in the sense that their elements are not geometric vectors, but they still follow the axioms of abstract vector spaces and can be operated on using the operations of addition and scalar multiplication.

5. What is the significance of abstract vector spaces in scientific research?

Abstract vector spaces are essential tools in scientific research, particularly in fields such as linear algebra, functional analysis, and physics. They provide a powerful framework for understanding and solving problems involving vectors, and their axioms and properties can be applied to a wide range of real-world applications. Additionally, the concept of abstract vector spaces has led to the development of more advanced mathematical theories and techniques, making them an important area of study in scientific research.

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