batballbat said:Homework Statement
[itex]S_1[/itex] and [itex]S_2[/itex] are subsets of a vector space. When is this:[itex]span(S_1 \cap S_2) = span(S_1) \cap span(S_2)[/itex] true? Prove it.Homework Equations
The Attempt at a Solution
conjecture: iff the two subsets are vector spaces.
It's easy to see that your guess is wrong. Let ##\{e_1,e_2\}## be the standard basis of ##\mathbb R^2##. Let ##S_1=\{e_1\}## and ##S_2=\{e_1,e_2\}##. We have $$\operatorname{span}S_1\cap\operatorname{span} S_2 =\operatorname{span}(S_1\cap S_2)$$ but neither ##S_1## nor ##S_2## is a subspace.batballbat said:Homework Statement
[tex] S_1 [/tex] and [tex] S_2 [/tex] are subsets of a vector space. When is this:[tex] span(S_1 \cap S_2) = span(S_1) \cap span(S_2) [/tex] true? Prove it.
Homework Equations
The Attempt at a Solution
conjecture: iff the two subsets are vector spaces.
A linear algebra span proof is a mathematical method used to prove that a vector is a linear combination of other vectors in a given vector space.
A linear algebra span proof is important because it helps to understand the relationship between different vectors and their span in a vector space. It also allows us to determine whether a vector can be expressed as a linear combination of other vectors in a given space.
A linear algebra span proof is performed by finding the coefficients that satisfy a linear combination of vectors that result in the given vector. This is typically done by setting up a system of equations and using techniques such as Gaussian elimination to solve for the coefficients.
A linear algebra span proof shows that a vector can be expressed as a linear combination of other vectors, while a linear independence proof shows that a set of vectors are not linearly dependent on each other. In other words, a linear algebra span proof focuses on the span of a given vector, while a linear independence proof focuses on the relationships between a set of vectors.
Understanding linear algebra span proofs is important in many areas of mathematics, physics, and engineering. It allows us to solve systems of linear equations, determine the dimension of a vector space, and perform other important calculations. It also provides a foundation for more advanced topics such as linear transformations and eigenvectors.