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 $v$.
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.
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.
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.
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.