Proving a Vector in $\Bbb R^2$ is of the Form $au+bv$

In summary, any point in $\mathbb{R}^2$ can be described as the coordinates of the tip of a vector, and any vector can be written as the sum of two perpendicular vectors, with one of the perpendicular vectors having a coefficient of $0$ depending on the orientation of the vector. This means that any vector in $\mathbb{R}^2$ can be written in the form $au + bv$, where $u$ and $v$ are two fixed vectors. This is true for general vectors $u = (u_1, u_2)$ and $v = (v_1, v_2)$ if they are not orthogonal, meaning that the determinant of the system in unknowns
  • #1
Niamh1
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0
Let $a, b \in \Bbb R$ and $u, v \in \Bbb R^2$, with $u = (0, 1)$ and $v = (1, 0)$. Show that every vector in $\Bbb R^2$ is of the form $au + bv$. Under what conditions is this true for general vectors $u = (u_1, u_2)$ and $v = (v_1, v_2)$?

No idea where to begin. Would appreciate any help.
 
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  • #2
Any point in $\mathbb{R}^2$ can be described as the coordinates of the tip of a vector. As any vector can be the hypotenuse of the triangle formed by such a vector and two perpendicular vectors (with either $a$ or $b$ being $0$ if our vector is vertical or horizontal), we can describe any vector with the vector sum given.

What if $<u_1,u_2>$ and $<v_1,v_2>$ are orthogonal? What if they are not orthogonal?

Does that help?
 
  • #3
Let $w=(w_1,w_2)$ be an arbitrary vector. With your notation, when can you find $a,\,b$ with $w=au+bv$. That is the system in unknowns a and b has a (unique) solution:
$$u_1a+v_1b=w_1$$
$$u_2a+v_2b=w_2$$

Since this is linear algebra you should know that the answer is that the determinant $u_1v_2-v_1u_2\neq0$.
 

FAQ: Proving a Vector in $\Bbb R^2$ is of the Form $au+bv$

What does it mean to prove a vector in $\Bbb R^2$ is of the form $au+bv$?

Proving a vector in $\Bbb R^2$ is of the form $au+bv$ means showing that the vector can be written as a linear combination of two other vectors, $u$ and $v$, with scalar coefficients $a$ and $b$, respectively. This is also known as the linear span or span of the vectors $u$ and $v$.

Why is it important to prove that a vector is of the form $au+bv$?

Proving that a vector is of the form $au+bv$ is important because it allows us to represent the vector in a simpler and more manageable way. By writing the vector as a combination of other vectors, we can better understand its properties and use it in various mathematical operations.

What are the steps involved in proving a vector is of the form $au+bv$?

The steps involved in proving a vector is of the form $au+bv$ may vary depending on the context, but generally involve showing that the vector can be written as a linear combination of two other vectors, and then solving for the scalar coefficients $a$ and $b$. This may involve using algebraic manipulation or geometric reasoning.

Can any vector in $\Bbb R^2$ be expressed as $au+bv$?

Yes, any vector in $\Bbb R^2$ can be expressed as $au+bv$, as long as the vectors $u$ and $v$ are linearly independent. This means that they are not scalar multiples of each other, and together they span the entire vector space $\Bbb R^2$. If the vectors are linearly dependent, there may be restrictions on the values of $a$ and $b$ that can be used.

How does proving a vector is of the form $au+bv$ relate to vector operations?

Proving that a vector is of the form $au+bv$ allows us to use the properties of vector operations to simplify calculations. For example, we can use the distributive property to expand the vector and perform operations on the individual components, rather than the entire vector as a whole. This can make calculations more efficient and easier to understand.

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